Matrix (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 25/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Matrix_(mathematics)


00:04:07 1 Definition
00:05:09 1.1 Size
00:06:01 2 Notation
00:09:53 3 Basic operations
00:10:30 3.1 Addition, scalar multiplication and transposition
00:10:48 3.2 Matrix multiplication
00:11:12 3.3 Row operations
00:11:24 3.4 Submatrix
00:12:20 4 Linear equations
00:12:42 5 Linear transformations
00:13:13 6 Square matrix
00:13:27 6.1 Main types
00:13:30 6.1.1 Diagonal and triangular matrix
00:16:29 6.1.2 Identity matrix
00:18:19 6.1.3 Symmetric or skew-symmetric matrix
00:18:51 6.1.4 Invertible matrix and its inverse
00:20:38 6.1.5 Definite matrix
00:23:04 6.1.6 Orthogonal matrix
00:25:23 6.2 Main operations
00:25:34 6.2.1 Trace
00:26:13 6.2.2 Determinant
00:26:46 6.2.3 Eigenvalues and eigenvectors
00:26:55 7 Computational aspects
00:27:23 8 Decomposition
00:28:55 9 Abstract algebraic aspects and generalizations
00:29:23 9.1 Matrices with more general entries
00:29:37 9.2 Relationship to linear maps
00:30:33 9.3 Matrix groups
00:30:49 9.4 Infinite matrices
00:31:12 9.5 Empty matrices
00:32:22 10 Applications
00:32:26 10.1 Graph theory
00:33:41 10.2 Analysis and geometry
00:34:38 10.3 Probability theory and statistics
00:34:46 10.4 Symmetries and transformations in physics
00:36:25 10.5 Linear combinations of quantum states
00:36:28 10.6 Normal modes
00:37:26 10.7 Geometrical optics
00:38:30 10.8 Electronics
00:39:29 11 History
00:39:59 11.1 Other historical usages of the word "matrix" in mathematics
00:42:04 12 See also
00:42:44 13 Notes
00:44:49 14 References
00:45:06 14.1 Physics references
00:45:37 14.2 Historical references
00:45:49 15 External links
00:46:40 Matrices with more general entries
00:48:31 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 square matrices, and thus need not be members of any ring; but their sizes must fulfil certain compatibility conditions.
00:49:15 Relationship to linear maps
00:49:25 Linear maps Rn → Rm are equivalent to m-by-n matrices, as described above. More generally, any linear map f: V → W between finite-dimensional vector spaces can be described by a matrix A
00:51:37 m composition of these maps is possible, and this gives rise to the matrix ring of n×n matrices representing the endomorphism ring of Rn.
00:51:49 Matrix groups
00:53:18 Infinite matrices
00:57:16 Empty matrices
00:58:17 Applications
01:00:13 Graph theory
01:01:06 Analysis and geometry
01:03:52 Probability theory and statistics
01:05:08 1, …, Nwhich 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.
01:05:31 Symmetries and transformations in physics
01:05:42 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.
01:06:47 Linear combinations of quantum states
01:06:58 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 k ...