Relation between AB and BA

linear-algebra eigenvalues-eigenvectors minimal-polynomials

Solution 1:

First note that there is an equivalent definition for characteristic value - a characteristic value of $A$ is a scalar $c$ such that the matrix $(A-cI)$ is NOT invertible.

For (i) - if $0$ is a characteristic value of $AB$ then $AB$ is not invertible $\Rightarrow BA$ is not invertible and hence $0$ is a characteristic value of $BA$. Let us assume that $c\neq 0$ is a characteristic value of $AB$. Then as you have shown $c$ is a characteristic value of $BA$ as long as $Bv\neq0$. Suppose $Bv=0\Rightarrow A(Bv)=0\Rightarrow cv=0$. Since $c\neq0$ we have $v=0$ which is a contradiction.

For (ii) and (iii) see Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?

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