- #1

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Why is it that if A is m×n-matrix and B is n×m matrices such that m<n, then AB is m×m and BA is n×n matrix. Then the following is true:

pAB(t) = t^(m-n)*pBA(t)

where pAB(t) and pBA(t) are characteristic polynomials of AB and BA

thanks

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- Thread starter pwhitey86
- Start date

- #1

- 5

- 0

Why is it that if A is m×n-matrix and B is n×m matrices such that m<n, then AB is m×m and BA is n×n matrix. Then the following is true:

pAB(t) = t^(m-n)*pBA(t)

where pAB(t) and pBA(t) are characteristic polynomials of AB and BA

thanks

- #2

AKG

Science Advisor

Homework Helper

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p

Why is this true? Well you'll have to fill in the details, but here are some facts to consider:

If [itex]\lambda[/itex] is a root of p

Also, since rank(A) is at most m, and likewise for B, rank(AB) and rank(BA) are at most m. But BA is nxn, so it has 0 as an eigenvalue with multiplicity at least n-m, accounting for the t

These are vague ideas, hopefully they lead you to a proof.

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