Minuri/sinhala-corpus-wikipedia huggingface dataset:
භෞතික විද්‍යාවෙහි, විද්‍යුත් පරිපථ, දෘශ්ටි විද්‍යාව සහ ක්වොන්ටම් යන්ත්‍ර  විද්‍යාව අධ්‍යයනය සඳහා න්‍යාස භාවිතාකරයි
පරිගණක ප්‍රස්තාරික ක්‍රමයෙහි, ද්විමාන තලයෙහි ත්‍රිමාන ප්‍රතිබිම්බ ප්‍රක්ෂේපනය කිරීමට සහ තාත්වික චලිතය පෙනෙන අයුරු නිර්මාණය කිරීමට න්‍යාස භාවිතාකරයි
බහු මාන සඳහා ව්‍යුත්පන්න සහ දර්ශක ඝාතීය යොදාගන්නවාසේම පෞරාණික විශ්ලේෂණාත්මක මත පිළිබඳ අවබෝධයක් ලබාගැනීමට න්‍යාස කලනය යොදාගනියි
අවුරුදු සිය ගණනක් තිස්සේ වස්තු විෂය වෙමින් වර්තමානය වන විට පර්යේෂණ මට්ටම දක්වාම විහිදී ගිය න්‍යාස ගණනය සඳහා කාර්යක්ෂම ඇල්ගොරිතම සංවර්ධනය කිරීම සංඛ්‍යාත්මක විශ්ලේෂණයේද ප්‍රධාන කොටසක් වී පවතී
සෛද්ධාන්තිකවත් ප්‍රායෝගිකවත්, න්‍යාස වියෝජන ක්‍රමය න්‍යාස ගණනය සරල කර දෙයි
එක් එක් න්‍යාස ආකෘති උදා
- විරල න්‍යාස, කර්ණය ආසන්න න්‍යාස, සඳහා ඈඳන ලද ඇල්ගොරිතම පරිමිත මුලාවයව ක්‍රමයේ හා වෙනත් න්‍යාස ගණනයන් ඉක්මන් කරවයි. ග්‍රහ තාරකා සිද්ධාන්ත වලදී සහ පරමාණුක සිද්ධාන්ත වලදී අපරිමිත න්‍යාස භාවිතා වෙයි
ශ්‍රිතයක ටේලර් ශ්‍රේණිය මත බලපාන ව්‍යුත්පන්න කාරකය නිරූපණය කරන න්‍යාසය ඒ සඳහා ඇති හොඳම නිදසුනයි
අර්ථදැක්වීම
න්‍යාස යනු ගණිතමය ප්‍රකාශනයන්හි සෘජුකෝණාස්‍රාකාර පිළියෙළ කිරීමක් වන අතර එමඟින් සංඛ්‍යාවල සරල බව ඇතිකරවිය හැකිය
නිදසුනක් ලෙස,

 
විකල්ප අංකනයක් ලෙස කොටු වරහන් වෙනුවට විශාල සුළු වරහන් යොදාගත හැක
න්‍යාසයක තිරස් පේළි සහ සිරස් තීරු අඩංගුවේ
න්‍යාසයක අඩංගු සංඛ්‍යා එහි මූලාවයව ලෙස හෝ ඇතුලත් කිරීම් ලෙස හඳුන්වනු ලබයි
තිරස් පේළි m ගණනකින් හා සිරස් තීරු n ගණනකින් යුතු න්‍යාසයක් m-by-n න්‍යාසයක් ලෙස හෝ m  n න්‍යාසයක් ලෙස එහි ප්‍රමාණය විශේෂයෙන් සඳහන් කරන අතර, m සහ n එහි මානවේ
ඉහත සඳහන් කර ඇත්තේ 4-by-3 න්‍යාසයකි
තිරස් පේළි එකක් (1  n) පමණක් ඇති න්‍යාස තිරස් පේළි දෛශික ලෙසද සිරස් තීරු එකක් (m  1) පමණක් ඇති න්‍යාස සිරස් පේළි දෛශික ලෙසද හඳුන්වනු ලබයි
න්‍යාසයක ඕනෑම තිරස් පේළියක් හෝ සිරස් තීරුවක් තිරස් පේළි දෛශිකය හෝ සිරස් තීරු දෛශිකය නිර්ණය කරන අතර, න්‍යාසයෙහි වෙනත් තිරස් පේළි හෝ සිරස් තීරු අනුපිළිවෙළින් ඉවත්කිරීම මඟින් ලබාගත හැකියි
නිසදසුනක් ලෙස, ඉහත A න්‍යාසයෙහි තුන්වන තිරස් පේළියේ තිරස් පේළි දෛශිකය

න්‍යාසයක තිරස් පේළිය හෝ සිරස් තීරුව අගයකට අර්ථපහදා දෙන විට, එය අනුරූප තිරස් පේළි දෛශිකයට හෝ සිරස් තීරු දෛශිකයට යොමුකරනු ලැබේ
උදාහරණයක් ලෙස න්‍යාසයක වෙනස් තිරස් පේළි දෙකක් සමාන බව කිව හැකියි, එහි අර්ථය ඒවායෙහි තිරස් පේළි දෛශිකය සමාන බවයි. සමහර අවස්ථාවලදී තිරස් පේළියක හෝ සිරස් තීරුවක අගය හරියටම අගයන් අනුපිළිවෙළින් (Rn හි මූලාවයවයකඇතුලත් කිරීම් තත්වික සංඛ්‍යා නම්) න්‍යාසයට වඩා වැඩියෙන් අර්ථපහදා දිය යුතුයි, උදාරණයක් ලෙස න්‍යාසයක තිරස් පේළි අනුරූප සිරස් තීරු වලට සමාන වන විට එය එහි තැන් පෙරළුම් න්‍යාසයවෙයි
බොහෝ සෙයින් මෙම ලිපිය තාත්වික සහ සංකීර්ණ න්‍යාස කේන්ද්‍ර කරගෙන ඇත, තව දුරටත් න්‍යාසයක මූලාවයව කෙසේදයත් අනුපිළිවෙළින් තාත්වික හෝ සංකීර්ණ විය හැකියි
බොහෝ සාමාන්‍ය ආකාරයේ ඇතුලත් කිරීම් පහත සාකච්ඡා කරනු ලබයි
අංකනය
යම් දෙසකට පැතිර පවත්නා බොහෝ පුළුල් වූ න්‍යාස විශේෂීකරණ අංකනයක් ඇත
සාමාන්‍යයෙන් ඉංග්‍රීසි ලොකු අකුරු භාවිතයෙන් න්‍යාසයක් දක්වනු ලබන අතර අනුරූප ඉංග්‍රීසි කුඩා අකුරු සමඟ යටිකුරු දර්ශක දෙකක් යෙදීමෙන් ඇතුලත් කිරීමක් ඉදිරිපත් කරනු ලබයි
ඊට අමතරව ඉංග්‍රීසි ලොකු අකුරු භාවිතයෙන් න්‍යාසයක් සංකේතවත් කරනු ලබයි, බොහෝ ලේඛකයන් විශේෂ මුද්‍රණ ශිල්පීය විලාසිතාභාවිතා කරයි, වෙනත් ගණිතමය දේවල් මඟින් න්‍යාස වල ඇති වෙනස හඳුනා ගැනීමට සුලබව සෘජු තද අකුරු (ඇල නැති) ආධාරකරනු ලබයි
විකල්ප අංකනයක් ලෙස සෘජු අකුරු සහිතව හෝ රහිතව විචල්‍ය නාමයට යටින් ඉරි දෙකක් ගැසීම සිදුකරනු ලබයි, (e
න්‍යාසයක i වෙනි තිරස් පේළියේ සහ j වෙනි සිරස් තීරුවේ ඇතුලත් කිරීම i,j වන ඇතුළත් කිරීම ලෙස සලකයි
නිදසුනක් ලෙස, ඉහත A න්‍යාසයේ (2,3) ඇතුළත් කිරිම 7 වේ
A නම් න්‍යාසයක (i, j) ඇතුළත් කිරීම බහුල වශයෙන් භාවිත කරනුයේ ai,j ලෙසිනි
A[i,j] හෝ Ai,j යනු මේ සඳහා භාවිතා කරනු ලබන වෙනත් සංකේත වේ
Sometimes a matrix is referred to by giving a formula for its (i,j)th entry, often with double parenthesis around the formula for the entry, for example, if the (i,j)th entry of A were given by aij, A would be denoted ((aij))
An asterisk is commonly used to refer to whole rows or columns in a matrix
For example, ai,∗ refers to the ith row of A, and a∗,j refers to the jth column of A
The set of all m-by-n matrices is denoted (m, n)
A common shorthand is
A = [ai,j]i = 1,
j = 1,
,n or more briefly A = [ai,j]mn
to define an m  n matrix A
Usually the entries ai,j are defined separately for all integers  and
They can however sometimes be given by one formula
for example the 3-by-4 matrix

can alternatively be specified by A = [i − j]i = 1,2,3
,4, or simply A = ((i-j)), where the size of the matrix is understood
Some programming languages start the numbering of rows and columns at zero, in which case the entries of an m-by-n matrix are indexed by  and
This article follows the more common convention in mathematical writing where enumeration starts from 1.

මූලික ක්‍රියාකාරකම්

There are a number of operations that can be applied to modify matrices called matrix addition, scalar multiplication and transposition
These form the basic techniques to deal with matrices
Familiar properties of numbers extend to these operations of matrices
for example, addition is commutative, i
, the matrix sum does not depend on the order of the summands
A + B = B + A
The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT
Finally, (AT)T = A
Row operations are ways to change matrices
There are three types of row operations
row switching, that is interchanging two rows of a matrix
row multiplication, multiplying all entries of a row by a non-zero constant
and finally row addition, which means adding a multiple of a row to another row
These row operations are used in a number of ways including solving linear equations and finding inverses
න්‍යාස ගුණකිරීම,රේඛීය සමීකරණ සහ රේඛීය පරිණාමණ

Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix
If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B
where 1 ≤ i ≤ m and 1 ≤ j ≤ p
For example, the underlined entry 1 in the product is calculated as 

Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity), and (A+B)C = AC+BC as well as C(A+B) = CA+CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined
The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and  Even if both products are defined, they need not be equal, i
, generally one has
AB ≠ BA,
i
, matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors
An example of two matrices not commuting with each other is
whereas

The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e
It is called identity matrix because multiplication with it leaves a matrix unchanged
MIn = ImM = M for any m-by-n matrix M
Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product
They arise in solving matrix equations such as the Sylvester equation
රේඛීය සමීකරණ

A particular case of matrix multiplication is tightly linked to linear equations
if x designates a column vector (i
, n1-matrix) of n variables x1, x2,
, xn, and A is an m-by-n matrix, then the matrix equation
Ax = b,
where b is some m1-column vector, is equivalent to the system of linear equations
A1,1x1 + A1,2x2 +
+ A1,nxn = b1
Am,1x1 + Am,2x2 +
+ Am,nxn = bm
This way, matrices can be used to compactly write and deal with multiple linear equations, i
, systems of linear equations
රේඛීය පරිණාමණ

Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps
A real m-by-n matrix A gives rise to a linear transformation Rn → Rm mapping each vector x in Rn to the (matrix) product Ax, which is a vector in Rm
Conversely, each linear transformation f
Rn → Rm arises from a unique m-by-n matrix A
explicitly, the  of A is the ith coordinate of f(ej), where ej = (0,
,0,1,0,
,0) is the unit vector with 1 in the jth position and 0 elsewhere
The matrix A is said to represent the linear map f, and A is called the transformation matrix of f
For example, the 22 matrix

can be viewed as the transform of the unit square into a parallelogram with vertices at , , , and
The parallelogram pictured at the right is obtained by multiplying A with each of the column vectors  and  in turn
These vectors define the vertices of the unit square
The following table shows a number of 2-by-2 matrices with the associated linear maps of R2. The blue original is mapped to the green grid and shapes, the origin (0,0) is marked with a black point
Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps
if a k-by-m matrix B represents another linear map g
Rm → Rk, then the composition  is represented by BA since
(g ∘ f)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x
The last equality follows from the above-mentioned associativity of matrix multiplication
The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors
Equivalently it is the dimension of the image of the linear map represented by A
The rank-nullity theorem states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix
සමචතුරස්‍රාකාර න්‍යාස
A square matrix is a matrix with the same number of rows and columns
An n-by-n matrix is known as a square matrix of order n
Any two square matrices of the same order can be added and multiplied
A square matrix A is called invertible or non-singular if there exists a matrix B such that
 AB = In
This is equivalent to BA = In
Moreover, if B exists, it is unique and is called the inverse matrix of A, denoted A−1.

The entries Ai,i form the main diagonal of a matrix
The trace, tr(A) of a square matrix A is the sum of its diagonal entries
While, as mentioned above, matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors
tr(AB) = tr(BA)
Also, the trace of a matrix is equal to that of its transpose, i
, tr(A) = tr(AT)
If all entries outside the main diagonal are zero, A is called a diagonal matrix
If only all entries above (below) the main diagonal are zero, A is called a lower triangular matrix (upper triangular matrix, respectively)
For example, if n = 3, they look like

 (diagonal),  (lower) and   (upper triangular matrix)
නිර්ණායකය

The determinant det(A) or |A| of a square matrix A is a number encoding certain properties of the matrix
A matrix is invertible if and only if its determinant is nonzero
Its absolute value equals the area (in R2) or volume (in R3) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map
the determinant is positive if and only if the orientation is preserved
The determinant of 2-by-2 matrices is given by

When the determinant is equal to one, then the matrix represents an equi-areal mapping
The determinant of 3-by-3 matrices involves 6 terms (rule of Sarrus)
The more lengthy Leibniz formula generalises these two formulae to all dimensions
The determinant of a product of square matrices equals the product of their determinants
det(AB) = det(A)  det(B)
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant
Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal
this provides a method to calculate the determinant of any matrix
Finally, the Laplace expansion expresses the determinant in terms of minors, i
, determinants of smaller matrices
This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula
Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables
අයිගන් අගයන් සහ අයිගන් දෛශික

A number λ and a non-zero vector v satisfying
Av = λv
are called an eigenvalue and an eigenvector of A, respectively
The number λ is an eigenvalue of an nn-matrix A if and only if A−λIn is not invertible, which is equivalent to

The function pA(t) = det(A−tI) is called the characteristic polynomial of A, its degree is n
Therefore pA(t) has at most n different roots, i
, eigenvalues of the matrix
They may be complex even if the entries of A are real
According to the Cayley–Hamilton theorem, pA(A) = 0, that is to say, the characteristic polynomial applied to the matrix itself yields the zero matrix
සමමිතිය
A square matrix A that is equal to its transpose, i
, A = AT, is a symmetric matrix
If instead, A was equal to the negative of its transpose, i
, A = −AT, then A is a skew-symmetric matrix
In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, i
, the transpose of the complex conjugate of A
By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis
, every vector is expressible as a linear combination of eigenvectors
In both cases, all eigenvalues are real
This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below
නිශ්චිත භාවය

A symmetric nn-matrix is called positive-definite (respectively negative-definite
indefinite), if for all nonzero vectors x ∈ Rn the associated quadratic form given by
Q(x) = xTAx
takes only positive values (respectively only negative values
both some negative and some positive values)
If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite)
hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite
A symmetric matrix is positive-definite if and only if all its eigenvalues are positive
The table at the right shows two possibilities for 2-by-2 matrices
Allowing as input two different vectors instead yields the bilinear form associated to A
BA (x, y) = xTAy
සංඛ්‍යාත්මක ආකාර
In addition to theoretical knowledge of properties of matrices and their relation to other fields, it is important for practical purposes to perform matrix calculations effectively and precisely
The domain studying these matters is called numerical linear algebra
As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stability
Many problems can be solved by both direct algorithms or iterative approaches
For example, finding eigenvectors can be done by finding a sequence of vectors xn converging to an eigenvector when n tends to infinity
Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, e
, multiplication of matrices
For example, calculating the matrix product of two n-by-n matrix using the definition given above needs n3 multiplications, since for any of the n2 entries of the product, n multiplications are necessary
The Strassen algorithm outperforms this "naive" algorithm
it needs only n2.807 multiplications
A refined approach also incorporates specific features of the computing devices
In many practical situations additional information about the matrices involved is known
An important case are sparse matrices, i
, matrices most of whose entries are zero
There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method
An algorithm is, roughly speaking, numerically stable, if little deviations (such as rounding errors) do not lead to big deviations in the result
For example, calculating the inverse of a matrix via Laplace's formula (Adj (A) denotes the adjugate matrix of A)
A−1 = Adj(A) / det(A)
may lead to significant rounding errors if the determinant of the matrix is very small
The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix' inverse
Although most computer languages are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC commands for matrices
Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices
න්‍යාස වියෝජන ක්‍රම

There are several methods to render matrices into a more easily accessible form
They are generally referred to as matrix transformation or matrix decomposition techniques
The interest of all these decomposition techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices
The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U)
Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution
Likewise, inverses of triangular matrices are algorithmically easier to calculate
The Gaussian elimination is a similar algorithm
it transforms any matrix to row echelon form
Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row
Singular value decomposition expresses any matrix A as a product UDV∗, where U and V are unitary matrices and D is a diagonal matrix
The eigendecomposition or diagonalization expresses A as a product VDV−1, where D is a diagonal matrix and V is a suitable invertible matrix
If A can be written in this form, it is called diagonalizable
More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal form, that is to say matrices whose only nonzero entries are the eigenvalues λ1 to λn of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right
Given the eigendecomposition, the nth power of A (i
, n-fold iterated matrix multiplication) can be calculated via
An = (VDV−1)n = VDV−1VDV−1.
VDV−1 = VDnV−1
and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead
This can be used to compute the matrix exponential eA, a need frequently arising in solving linear differential equations, matrix logarithms and square roots of matrices
To avoid numerically ill-conditioned situations, further algorithms such as the Schur decomposition can be employed
වීජීය සංක්ෂේපන ආකාරය සහ සාධාරණීකරණය
Matrices can be generalized in different ways
Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps
It is possible to consider matrices with infinitely many columns and rows
Another extension are tensors, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional array of numbers
Matrices, subject to certain requirements tend to form groups known as matrix groups
න්‍යාස සමඟ බොහෝ සාමාන්‍ය ඇතුලත් කිරීම්
This article focuses on matrices whose entries are real or complex numbers
However, matrices can be considered with much more general types of entries than real or complex numbers
As a first step of generalization, any field, i
, a set where addition, subtraction, multiplication and division operations are defined and well-behaved, may be used instead of R or C, for example rational numbers or finite fields
For example, coding theory makes use of matrices over finite fields
Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a larger field than that of the coefficients of the matrix
for instance they may be complex in case of a matrix with real entries
The possibility to reinterpret the entries of a matrix as elements of a larger field (e
, to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues
Alternatively one can consider only matrices with entries in an algebraically closed field, such as C, from the outset
More generally, abstract algebra makes great use of matrices with entries in a ring R
Rings are a more general notion than fields in that no division operation exists
The very same addition and multiplication operations of matrices extend to this setting, too
The set M(n, R) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn
If the ring R is commutative, i
, its multiplication is commutative, then M(n, R) is a unitary noncommutative (unless n = 1) associative algebra over R
The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula
such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible
Matrices over superrings are called supermatrices
Matrices do not always have all their entries in the same ring – or even in any ring at all
One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices
The entries need not be quadratic matrices, and thus need not be members of any ordinary ring
but their sizes must fulfil certain compatibility conditions
රේඛීය සිතියම් වලට ඇති සබැඳියාව
Linear maps Rn → Rm are equivalent to m-by-n matrices, as described above
More generally, any linear map  between finite-dimensional vector spaces can be described by a matrix A = (aij), after choosing bases v1,
, vn of V, and w1,
, wm of W (so n is the dimension of V and m is the dimension of W), which is such that

In other words, column j of A expresses the image of vj in terms of the basis vectors wi of W
thus this relation uniquely determines the entries of the matrix A
Note that the matrix depends on the choice of the bases
different choices of bases give rise to different, but equivalent matrices
Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the transpose of the linear map given by A, with respect to the dual bases
More generally, the set of mn matrices can be used to represent the R-linear maps between the free modules Rm and Rn for an arbitrary ring R with unity
When n = m composition of these maps is possible, and this gives rise to the matrix ring of nn matrices representing the endomorphism ring of Rn
න්‍යාස කාණ්ඩ

A group is a mathematical structure consisting of a set of objects together with a binary operation, i
, an operation combining any two objects to a third, subject to certain requirements
A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group
Since in a group every element has to be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups
Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups
For example, matrices with a given size and with a determinant of 1 form a subgroup of (i
, a smaller group contained in) their general linear group, called a special linear group
Orthogonal matrices, determined by the condition
MTM = I,
form the orthogonal group
They are called orthogonal since the associated linear transformations of Rn preserve angles in the sense that the scalar product of two vectors is unchanged after applying M to them
(Mv)  (Mw) = v  w
Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group
General groups can be studied using matrix groups, which are comparatively well-understood, by means of representation theory
අපරිමිත න්‍යාස
It is also possible to consider matrices with infinitely many rows and/or columns even if, being infinite objects, one cannot write down such matrices explicitly
All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers)
The basic operations of addition, subtraction, scalar multiplication and transposition can still be defined without problem
however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general
If R is any ring with unity, then the ring of endomorphisms of  as a right R module is isomorphic to the ring of column finite matrices  whose entries are indexed by , and whose columns each contain only finitely many nonzero entries
The endomorphisms of M considered as a left R module result in an analogous object, the row finite matrices  whose rows each only have finitely many nonzero entries
If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason
For a matrix A to describe a linear map f
V→W, bases for both spaces must have been chosen
recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector v of coefficients, only finitely many entries vi are nonzero
Now the columns of A describe the images by f of individual basis vectors of V in the basis of W, which is only meaningful if these columns have only finitely many nonzero entries
There is no restriction on the rows of A however
in the product Av there are only finitely many nonzero coefficients of v involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined
Moreover this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns do
One also sees that products of two matrices of the given type is well defined (provided as usual that the column-index and row-index sets match), is again of the same type, and corresponds to the composition of linear maps
If R is a normed ring, then the condition of row or column finiteness can be relaxed
With the norm in place, absolutely convergent series can be used instead of finite sums
For example, the matrices whose column sums are absolutely convergent sequences form a ring
Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring
In that vein, infinite matrices can also be used to describe operators on Hilbert spaces, where convergence and continuity questions arise, which again results in certain constraints that have to be imposed
However, the explicit point of view of matrices tends to obfuscate the matter, and the abstract and more powerful tools of functional analysis can be used instead
ශූන්‍ය න්‍යාස
An empty matrix is a matrix in which the number of rows or columns (or both) is zero
Empty matrices help dealing with maps involving the zero vector space
For example, if A is a 3-by-0 matrix A and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix
There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them
The determinant of the 0-by-0 matrix is 1 as follows from regarding the empty product occurring in the Leibniz formula for the determinant as 1. This value is also consistent with the fact that the identity map from any finite dimensional space to itself has determinant 1, a fact that is often used as a part of the characterization of determinants
භාවිතය
There are numerous applications of matrices, both in mathematics and other sciences
Some of them merely take advantage of the compact representation of a set of numbers in a matrix
For example, in game theory and economics, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose
Text mining and automated thesaurus compilation makes use of document-term matrices such as tf-idf to track frequencies of certain words in several documents
Complex numbers can be represented by particular real 2-by-2 matrices via

under which addition and multiplication of complex numbers and matrices correspond to each other
For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above
A similar interpretation is possible for quaternions
Early encryption techniques such as the Hill cipher also used matrices
However, due to the linear nature of matrices, these codes are comparatively easy to break
Computer graphics uses matrices both to represent objects and to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation
Matrices over a polynomial ring are important in the study of control theory
Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy
Examples are the overlap matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular orbitals of the Hartree–Fock method
ප්‍රස්තාරික සිද්ධාන්තය

The adjacency matrix of a finite graph is a basic notion of graph theory
It saves which vertices of the graph are connected by an edge
Matrices containing just two different values (0 and 1 meaning for example "yes" and "no") are called logical matrices
The distance (or cost) matrix contains information about distances of the edges
These concepts can be applied to websites connected hyperlinks or cities connected by roads etc
, in which case (unless the road network is extremely dense) the matrices tend to be sparse, i
, contain few nonzero entries
Therefore, specifically tailored matrix algorithms can be used in network theory
විශ්ලේෂණය සහ ජ්‍යාමිතිය
The Hessian matrix of a differentiable function
Rn → R consists of the second derivatives of  with respect to the several coordinate directions, i
It encodes information about the local growth behaviour of the function
given a critical point x = (x1,
, xn), i
, a point where the first partial derivatives  of  vanish, the function has a local minimum if the Hessian matrix is positive definite
Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above)
Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f
Rn → Rm
If f1,
, fm denote the components of f, then the Jacobi matrix is defined as 

If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theorem
Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation
For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question
The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems
It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation
සම්භාවිතා සිද්ධාන්ත සහ සංඛ්‍යානය

Stochastic matrices are square matrices whose rows are probability vectors, i
, whose entries sum up to one
Stochastic matrices are used to define Markov chains with finitely many states
A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row
Properties of the Markov chain like absorbing states, i
, states that any particle attains eventually, can be read off the eigenvectors of the transition matrices
Statistics also makes use of matrices in many different forms
Descriptive statistics is concerned with describing data sets, which can often be represented in matrix form, by reducing the amount of data
The covariance matrix encodes the mutual variance of several random variables
Another technique using matrices are linear least squares, a method that approximates a finite set of pairs (x1, y1), (x2, y2),
, (xN, yN), by a linear function
yi ≈ axi + b, i = 1,
, N
which can be formulated in terms of matrices, related to the singular value decomposition of matrices
Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution
Beyond probability theory, they are applied in domains ranging from number theory to physics
සමමිති සහ භෞතික විද්‍යාව තුළ පරිණාමණ

Linear transformations and the associated symmetries play a key role in modern physics
For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group
Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions, which behave as spinors
For the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3)
for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamics
The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses
ක්වොන්ටම් ගති ස්වභාවයේ රේඛීය සංයෝජන
The first model of quantum mechanics (Heisenberg, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states
This is also referred to as matrix mechanics
One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates
Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics
Collision reactions such as occur in particle accelerators, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states
The linear combination is given by a matrix known as the S-matrix, which encodes all information about the possible interactions between particles
සාමාන්‍ය ක්‍රම
A general application of matrices in physics is to the description of linearly coupled harmonic systems
The equations of motion of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions
The best way to obtain solutions is to determine the system's eigenvectors, its normal modes, by diagonalizing the matrix equation
Techniques like this are crucial when it comes to the internal dynamics of molecules
the internal vibrations of systems consisting of mutually bound component atoms
They are also needed for describing mechanical vibrations, and oscillations in electrical circuits
ජ්‍යාමිතික දෘශ්ටි විද්‍යාව
Geometrical optics provides further matrix applications
In this approximative theory, the wave nature of light is neglected
The result is a model in which light rays are indeed geometrical rays
If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix
the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element
Actually, there are two kinds of matrices, viz
a refraction matrix describing the refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies
The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components' matrices
ඉලෙක්ට්‍රෝනික විද්‍යාව
Traditional mesh analysis in electronics leads to a system of linear equations that can be described with a matrix
The behaviour of many electronic components can be described using matrices
Let A be a 2-dimensional vector with the component's input voltage v1 and input current i1 as its elements, and let B be a 2-dimensional vector with the component's output voltage v2 and output current i2 as its elements
Then the behaviour of the electronic component can be described by  B = H  A, where H is a 2 x 2 matrix containing one impedance element (h12), one admittance element (h21) and two dimensionless elements (h11 and h22)
Calculating a circuit now reduces to multiplying matrices
ඉතිහාසය
Matrices have a long history of application in solving linear equations
The Chinese text The Nine Chapters on the Mathematical Art (Jiu Zhang Suan Shu), from between 300 BC and AD 200, is the first example of the use of matrix methods to solve simultaneous equations, including the concept of determinants, over 1000 years before its publication by the Japanese mathematician Seki in 1683 and the German mathematician Leibniz in 1693. Cramer presented his rule in 1750.

Early matrix theory emphasized determinants more strongly than matrices and an independent matrix concept akin to the modern notion emerged only in 1858, with Cayley's Memoir on the theory of matrices
The term "matrix" (Latin for "womb", derived from mater—mother) was coined by Sylvester, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows
In a 1851 paper, Sylvester explains
I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent
The study of determinants sprang from several sources
Number-theoretical problems led Gauss to relate coefficients of quadratic forms, i
, expressions such as  and linear maps in three dimensions to matrices
Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are non-commutative
Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = [ai,j] the following
replace the powers ajk by ajk in the polynomial

where Π denotes the product of the indicated terms
He also showed, in 1829, that the eigenvalues of symmetric matrices are real
Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or infinitesimal) level, see above
Kronecker's Vorlesungen über die Theorie der Determinanten and Weierstrass' Zur Determinantentheorie, both published in 1903, first treated determinants axiomatically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy
At that point, determinants were firmly established
Many theorems were first established for small matrices only, for example the Cayley–Hamilton theorem was proved for 22 matrices by Cayley in the aforementioned memoir, and by Hamilton for 44 matrices
Frobenius, working on bilinear forms, generalized the theorem to all dimensions (1898)
Also at the end of the 19th century the Gauss–Jordan elimination (generalizing a special case now known as Gauss elimination) was established by Jordan
In the early 20th century, matrices attained a central role in linear algebra
partially due to their use in classification of the hypercomplex number systems of the previous century
The inception of matrix mechanics by Heisenberg, Born and Jordan led to studying matrices with infinitely many rows and columns
Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as linear operators on Hilbert spaces, which, very roughly speaking, correspond to Euclidean space, but with an infinity of independent directions
ලෝක ඉතිහාසයේ ගණිතය තුළ න්‍යසයන්හි වෙනත් භාවිත
The word has been used in unusual ways by at least two authors of historical importance
Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) use the word matrix in the context of their Axiom of reducibility
They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the “bottom” (0 order) the function is identical to its extension
“Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables
Then any possible function other than a matrix is derived from a matrix by means of generalization, i
, by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined”
For example a function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, e
, y, by “considering” the function for all possible values of “individuals” ai substituted in place of variable x
And then the resulting collection of functions of the single variable y, i
, ∀ai
Φ(ai, y), can be reduced to a “matrix” of values by “considering” the function for all possible values of “individuals” bi substituted in place of variable y
∀bj∀ai
Φ(ai, bj)
Alfred Tarski in his 1946 Introduction to Logic'' used the word “matrix” synonymously with the notion of truth table as used in mathematical logic
See also

 Algebraic multiplicity
 Geometric multiplicity
 Gram-Schmidt process
 List of matrices
 Matrix calculus
 Periodic matrix set
 Tensor

සටහන්

නිර්දේශ

භෞතික විද්‍යා නිර්දේශ

පෞරාණික නිර්දේශ
 , reprint of the 1907 original edition

භාහිර ඈඳුම්

 History
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Matrices and Linear Algebra on the Earliest Uses Pages
Earliest Uses of Symbols for Matrices and Vectors

 Online books
 
 
 

 Online matrix calculators
 
 
 
 
 

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Abstract algebra
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ගණිතය
හරිත පරිගණක යනු නිවැරදිව පරිගණක සම්පත් පරිහරණය අධ්‍යයනය හා ප්‍රායෝගික පුහුණුවයි. මෙවැනි වැඩසටහනක ප්‍රාථමික අභිප්‍රාය වන්නේ ත්‍රිත්ව පදනම් පෙළ ගණනය කිරීම හා සංවිධානාත්මක (සහ සමාජගත) සාර්ථකත්වය මැනීමට විහිදුනු වර්ණාවලියක අගයන් හා උපමානයි
මෙහි පරමාර්ථ හරිත රසායන විද්‍යාවට සමානය
අපද්‍රව සහිත අමුද්‍රව්‍ය භාවිතය අවම කිරීම , ශක්ති කාර්යක්ෂමතාව නිෂ්පාදනයේ ජීව කාලය තුළ වැඩි කිරීම හා අභාවයට පත් නිෂ්පාදන හා කර්මාන්තශාලා අපද්‍රව්‍යවල ප්‍රතිචක්‍රීකරණ හැකියාව හෝ ක්ෂ්‍රද්‍ර ජීවීන් නිසා විනාශයට පත්වන භාවය ඉහළ දමයි
නුතන තොරතුරු  තාක්ෂණ පද්ධති සංකීර්ණ මිනිස් සම්මිශ්‍රණයකින් මෙන්ම , ජාල හා දෘඩාංග මත රදා පවතී
එනම් හරිත පරිගණක මුල පිරිය යු‍ත්තේ ස්වභාව ධර්මයා හා විධිමත් , හා වැඩිවන බාල කළ ප්‍රශ්න වෙත යොමුව, විසදුමක කොටස් ලෙස භාවිතා කරන්නාගේ සෑහීම , නැවත පිළියෙල කිරීමේ කළමනාකාරීත්වය , යාමනය කළ අනුකූලතාව , ඉලෙක්ට්‍රෝන අපද්‍රව්‍ය බැහැර කිරීම , විදුලි සංදේශනය , සර්වර් සම්පත් වල තාත්ත්වීකරණය , ශක්ති භාවිතය , දුර්වල අනුග්‍රාහක විසදුම් , හා ආයෝජනය නැවත ලැබීම ආදී දෑ අඩංගු විය හැක
පරිගණක
පරිගණක විද්‍යාව
රෙසිඩන්ට් වෛරසවල ද , නන් රෙසිඩන්ට් වෛරසවලට සමාන පුණර්ජනාත්මක මොඩියුලයක් අඩංගු වේ
කෙසේ නමුත් මෙම මොඩියුලය සෙවුම් මොඩියුලයක් මගින් කැඳවනු නොලැබේ
ඒ වෙනුවට වෛරසය ක්‍රියාත්මක කළ විට එය පුණර්ජනාත්මක මොඩියුලය මතකයට ඇතුල් කර මෙහෙයුම් පද්ධතිය ‍යම් කාර්යයක් සිදු කරන සෑම අවස්ථාවක දීම එය ක්‍රියාත්මක කරවීමට වග බලා ගනී
උදාහරණයක් ලෙස මෙහෙයුම් පද්ධතිය ගොණුවක් විවෘත කරන සෑම අවස්ථාවකම පුණර්ජනාත්මක මොඩියුලය ක්‍රියාත්මක විය හැකිය
මේ නිසා වෛරසයක් පරිගණකයේ ක්‍රියාත්මක වන සෑම සුදුසු ක්‍රම‍ෙල්ඛයක්ම ආසාදනය කරයි
රෙසිඩන්‍ට් වෛරස සමහර විටෙක වේගවත් ආසාදනයන් හා මන්දගාමී ආසාදනයන් යනුවෙන් උප කොටස් ‍දෙකකට බෙදේ
වේගවත් ආසාදනයන් නිර්මාණය කර ඇත්තේ හැකිතාක් ගොණු ආසාදනය කිරීමටය
උදාහරණ ලෙස වේගවත් ආසාදනයන්ට ඇතුල්වන සියලු විභව සත්කාරක ගොණු ආසාදනය කළ හැක
මෙය ප්‍රතිවෛරස මෘදුකාංගවලට විශේෂ ප්‍රශ්නයක් ඇති කරයි
ප්‍රතිවෛරස මෘදුකාංග පුළුල් පද්ධති පරිලෝකනයක් සිදුකරන සෑම අවස්ථාවකදීම සෑම විභව සත්කාරක ගොණු සියල්ලටම ඇතුල් වේ
එවැනි වෛරසයක් මතකයේ ඇති බව වෛරස ස්කෑනරය තුළට ඇතුල් වී පරිලෝකනය කළ සියලු ගොණු ආසාදනය කළ හැක
වේගවත් ආසාදකයන් පැතිරීමේදී ඔවුන්ගේ වේගවත් ආසාදන සීඝ්‍රතාව ආධාර කර ගනී
මෙම ක්‍රමයේ අවාසිය වන්නේ ගොණු විශාල ප්‍රමාණයක් ආසාදනය කිරීම නිසා පහසුවෙන් අනාවරණය විය හැකි වීමයි
මෙම වෛරස පරිගණකවල වේගය අඩු කරන නිසා හෝ ප්‍රතිවෛරස් මෘදුකාංගවල අවධානයට යොමු වන ක්‍රියා සිදු කිරීම නිසා අනාවරණය පහසුවෙයි
මන්දගාමී ආසාදකයන් සැලසුම් කර ඇත්තේ සත්කාරක ගොණු සීඝ්‍රව ආසාදනය කිරීමට නොවේ
උදාහරණ ලෙස සමහරක් මන්දගාමී වෛරසවලට ගොණු පිටපත් කිරීමේදී ආසාදනය කළ හැක
මන්දගාමී වෛරස ඒවායේ ක්‍රියාකාරිත්වය සීමා කිරීමෙන් අනාවරණය වළක්වා ගැනීමට සැලසුම් කර ඇත
එය පරිගණකයේ අවධානයට බදුන්වන ලෙස මන්දගාමී නොකරන අතර ප්‍රතිවෛරස මෘදුකාංගවල අවධානය බඳුන් වන සැක කටයුතු හැසිරීම් දක්වන්නේ ඉතා අල්ප වශයෙනි
කෙසේ නමුත් මන්දගාමී ආසාදන ක්‍රියා පිළිවෙල සාර්ථක ලෙස නොසැ‍ලකේ.

References 
http
org/wiki/Computer_virus#Resident_viruses
ගණිතයෙහි දී ප්‍රකෘති සංඛ්‍යාවක් (මේවා ගණින සංඛ්‍යා සංඛ්‍යා ලෙසද හඳුන්වයි.) යනු { 1,2,3} අවයව සහිත සංඛ්‍යා කුලකයේ අවයවත් (ධන නිඛිල) හෝ {0,1,2,3.} අවයව සහිත සංඛ්‍යා (ඍණ නොවන නිඛිල ) කුලකයක් අවයවයක් වේ‍
මේ අතරින් මුල් ආකාරය සංඛ්‍යා වාදයේදී භාවිතාවන අතර පසු ආකාරය ගණිතමය තර්කනය කුලකවාදය හා පරිගණක විද්‍යාවේදී භාවිතා වේ
වඩාත් විධිමත් අර්ථකතනයක් පසුව දැක්වේ
ප්‍රකෘති සංඛ්‍යාවල ප්‍රධාන ප්‍රයෝජන දෙකකි
ඒවා ගණන් කිරීමට භාවිතා කළ හැක
(උදාහරණ - මේසය උඩ ඇපල් ගෙඩි තුනක් ඇත
) ඒවා පටිපාතගත කිරීම්වලට භාවිතා කරයි
(මෙය රටේ ඇති තුන්වැනි විශාලම නගරයයි
)

භාජ්‍යතාව ආදී ප්‍රකෘති සංඛ්‍යාවල ගුණන සංඛ්‍යාවාදය යටතේ ප්‍රථමක සංඛ්‍යා ව්‍යාප්තිය ආදී ලෙසට අධ්‍යයනයට ලක් වේ
රැම්සේ වාදය වැනි ගණන් කිරීම හා සම්බන්ධ ගැටළු සමායෝජන විද්‍යාව යටතේ අධ්‍යයනය කෙ‍රේ
සංඛ්‍යා
ගණිතය
ස්වාභාවික සංඛ්‍යා
ගණිතයේ දී ඔප්පු කිරීම මගින් අදහස් වන්නේ යම් ගණිතමය ප්‍රකාශනයක් ආදර්ශනය කිරීම තුළින් එය ගණිත පසුබිමේ ඇති යම් යම් තත්වයනට අනුකූලව සත්‍ය වන බව පෙන්වීමයි
එහිදී ඔප්පු කිරීම යනු අනුභවික නොවන තාර්කික කටයුත්තකි
ඔප්පු කිරීමකදී එම ගණිතමය ප්‍රකාශනය එය ආදර්ශනය කරන සෑම විටම එක අවස්ථාවක නොඇර එය සත්‍ය විය යුතුය
ඔප්පු නොවන ප්‍රශ්තුතයක් (proposition) සැම විටම ඌනණයක් (conjecture) ලෙස හැඳින්වේ
ඔප්පු කිරීමට තාර්කික බව ඇතුළත් වුවද එයට උභයාර්ථය ගෙන දීමට සැමවිටම ස්වාභාවික භාෂාව (natural language) මිශ්‍ර වීමක් ද වේ
ගණිතයේ ලියවී ඇති ඔප්පු කිරීම් වලින් වැඩි ප්‍රමාණයක් සාමාන්‍ය තර්ක ශාත්‍රිය ඔප්පු කිරීම් වේ
ඇත්ත වශයෙන්ම විධිමත් ඔප්පු කිරීම් සාධනය කිරීමේ සිද්ධාන්තවලට අයත් වේ
විධිමත් ඔප්පු කිරීම් හා සාමාන්‍ය ඔප්පු කිරීම් අතර පවතින වෙනස තුලින් නූතන හා පැරණි ගණිතයේ ගණිතමය ව්‍යවහාරයන්, ගණිතයේ අර්ධ අනුභූතිවාදය (quasi-empiricism) හා ජන ගණිතයයැයි කියාගන්නා අංශ පිළිබද පර්යේෂණ වර්ධනය වී ඇත