A n &#x00D7;<!-- × --> n matrix A is a linear transformation from the space of

aligass2004yi

aligass2004yi

Answered question

2022-06-15

A n × n matrix A is a linear transformation from the space of n × n matrices to itself. What is the n 2 × n 2 matrix corresponding to this transformation? If we knew the nullity and rank of A, how could we find the nullity and rank for this transformation?

Answer & Explanation

knolsaadme

knolsaadme

Beginner2022-06-16Added 16 answers

Let R n × n denote the set of n by n matrices with real entries. A matrix A R n × n specifices a linear map ϕ A : R n × n R n × n through matrix matrix multipliciation, i.e
X R n × n : ϕ A ( X ) = A X .
By definition, ϕ A ( X ) = A X can be computed by computing A x i for each column vector x i of X . If we where to stack the columns of X on top of each other, forming the gigantic vector x ~ R n 2 given by x ~ = ( x 1 T , x 2 T , , x n T ) T
and wanted to obtain a a similar representation of Y = A X, this could be achieved through the matrix vector multiplication
y ~ = A ~ x ~
where A ~ R n 2 × n 2 is obtained by using n copies of A as diagonal blocks in a gigantic matrix, i.e.
A ~ = ( A A A )
This form reveals that if A has rank r, then A ~ has rank n r.
There is good mathematical notation for the objects that I have used here. Normally, one writes
x ~ = vec ( X )
and
A ~ = I n A
where I n is the n by n identity matrix and denotes the Kronecker product.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Linear algebra

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?