Linear Algebra for QC
By: Rahulp Cs
Tags: Mathematics, Quantum Computing, Quantum Mechanics, Linear Algebra

a vector space is a set of vectors which follow certain axioms:

axioms of vector space.

the operations a vector space consists are vector addition and scalar multiplication(remember this ,since it will make it easier to understand Hilbert space later).

linear independence: lets consider a set of vectors a,b,c,…,k

now lets consider the set of all linear combinations of these vectors . c1a+c2b+c3c+…..ckk , where c1,c2,c3…,ck are scalars.

if any of these linear combinations results in a 0 vector and if all the scalars are also 0 then such a collection of vectors are linearly independent.

linearly dependent: if the linear combination of the above vectors is 0 vector when all the scalars are not 0 then such a collection of vectors is said to linearly dependent. we can express any one vector from the collection as a linear combination of other vectors.

span:

span of a set of vectors

basis: a basis for a vector space is the minimum no. of vectors needed to span that vector space. naturally this means that the basis set should be linearly independent. for an arbitrary vector space there can be infinitely many basis. each vector in the vector space can be represented uniquely in a given basis. all basis have the same no. of vectors.

To span a vector space we need at least the no. of vectors as in the basis, also the no. of vectors that are linearly independent should be less than the no. of vectors in the basis, hence the no. of vectors in basis of a vector space is a extremely important parameter and is called its dimension.

Hilbert space: A Hilbert space comes with additional properties and operations than a vector space but for all intents and purposes here we will consider only one of them, i.e a hilbert space is a vector space equipped with an inner product.

if H is a hilbert space and a,b are 2 vectors ∈ H, then <a|b>(inner product of a and b written in dirac notation) operation is defined and ∈ H .

Eigenvalues and Eigen vectors: Typically when a matrix is multiplied by a vector ,it is both scaled and rotated, but when a matrix is multiplied by its Eigenvector it is only scaled. let A be a matrix and a be its eigenvector than Aa is equivalent to λa where λ is a scalar. This lambda is called the corresponding eigen value. for Aa to be defined the no. of columns of A should be equal to the no. of rows of a , and the no. of rows of A should be equal to the no. of rows of a since Aa=λa ,hence A is a square matrix. For a matrix to have at least one eigenvector the matrix needs to be square ,also a given matrix can have infinitely many eigenvectors.

solving for eigenvalues and eigenvectors: Aa=λIa

Aa-λIa=0

(A-λI)a=0

B=A-λI

Ba=0 — — — — — (1)

the above equation has 2 solutions:

trivial where B is invertible => (B^-1)Ba=(B^-1)0 =>a=0 non-trivial where B is not invertible i.e B is singular i.e det(B)=0

det(A-λI)=0

when we solve for the above equation we get a canonical equation which looks like this:

(λ-λ1)(λ-λ2)(λ-λ3)….(λ-λn)=0

where λ1,λ2,….λn are the roots of canonical equation and the possible eigenvalues(not necessarily distinct). the no. of roots will always be equal to the dimension of the matrix. we can solve for the eigenvectors by plugging the eigenvalues in eqn 1.

Thanks a lot!