Using these, we express the given expression in terms of only $A$ and $P$.
On one hand, we have
On the other hand we have
Thus these are equal.
This completes the proof.
Did I make a mistake? Isn’t it too simple to be a qualifying exam problem?
Inequality Regarding Ranks of Matrices
Let $A$ be an $n \times n$ matrix over a field $K$. Prove that
where $\rk(B)$ denotes the rank of a matrix $B$.
(University of California, Berkeley, Qualifying Exam)
Regard the matrix as a linear transformation $A: […]
Linear Dependent/Independent Vectors of Polynomials
Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent?
(a) At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ […]