An exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...
The Cayley-Hamilton theorem is equivalent to: Let $R$ be a ring and let $M_n(R)$ be $n\times n$ matrices over $R$. Then the minimal polynomial of $A \in M_n(R)$ over $R$ divides the characteristic ...
I'm looking for a proof (using basic tools : definition of the characteristic polynomial and its basic properties) of the following fact : The roots of the characteristic polynomial of a symmetric ...
A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
There are two statement about a matrix under rank-one updates that I would be grateful if you give me some insightful proofs. Suppose $A$ be a nonsingular $n \times n$ matrix and ...
I'll put the question first, and then the background, because I'm not sure that the background is necessary to answer the question: I have a geometric proof, but is there an elegant algebraic proof ...