matrix multiplication

Question

let's say we have the following matrix multiplication:

A = B * C

where A,B,C are square (NxN) matrices.

how can i get B from the above statement?

is it:

B = C^-1 * A

or

B = A * C^-1

or something else???

(where C^-1 is the inverse matrix.)

thanks!

Answer 1

I'm a little rusty on my algebra, but just looking at it:

A = B * C

A * C(inv) = B * C * C(inv)

A * C(inv) = B

B = A * C(inv)

Answer 2

Close, Larry, but remember that matrix multiplication is NOT associative.

A = B * C

A* C(right C-inverse) = B * C * C(right inverse)

A * C(right inverse) = B

B = A * C(right inverse)

Answer 3

You're advised not to calculate the inverse of C using Gauss' method.

You'd use LU decomposition to reduce C to lower and upper triangular form, then forward-back substitution to calculate the solution.

We're assuming, of course, that C has an inverse. It may not. If C is square, it can't have a zero determinant.

If the number of columns exceeds the number of rows, multiply both sides by C(transpose) to get a square matrix and and solve that to get a least squares solution.

If the number of rows exceeds the number of columns, use Singular Value Decomposition instead of LU to get the best solution you can. It'll tell you the size of C's null space.

Answer 4

B = A * C^-1

However, C^-1 does not necessarily exist. It might be considered overkill, but please investigate Singular Value Decomposition (SVD) for a safe way to find a matrix (pseudo)inverse.

Answer 5

Here is a link with source code on solving matrix equations and matrix inversion using LU decomposition.