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Matrix Product Poser (Posted on 2016-03-19) Difficulty: 3 of 5
A is a 3x2 matrix and B is a 2x3 matrix, and:
AB = 
⌈ 8  2 -2⌉
| 2  5  4|
⌊-2  4  5⌋
Find the product BA.

No Solution Yet Submitted by K Sengupta    
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Family of Solutions Comment 3 of 3 |
In my first solution I decomposed the given matrix M into M = P*A'*B'*P^-1 to get A = P*A' and B = B'*P^-1. This yielded a final product (call it N) which is just a diagonal matrix.

It turns out with an adjustment I can generate an entire family of matricies N.  Insert Q*Q^-1 in the middle of M = P*A'*B'*P^-1 to get M = P*A'*Q*Q^-1*B'*P^-1.

Then A = P*A'*Q and B = Q^-1*B'*P^-1, which makes B*A = Q^-1*N*Q.  Since N is a diagonal matrix then this expression for B*A is a diagonalization of another matrix.  So in general for any given M, the set of all products B*A is the set of all matricies whose eigenvalues are the same as the nonzero eigenvalues of M.

This also applies to the orthogonal diagonalization in my second solution - just make Q an orthogonal matrix.

  Posted by Brian Smith on 2016-03-21 23:16:52
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