3
votes
1answer
27 views

Determinant of a matrix with symmetric positive definite block

In reviewing linear algebra for an exam, I encountered the following problem: Let $A \in \mathbb{R}^{n\times n}$ be symmetric positive definite. If $x$ is any nonzero vector, show that $$ ...
1
vote
0answers
39 views

How to prove that the inverse of a persymmetric matrix is also persymmetric?

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 ...
0
votes
1answer
86 views

Is there a simpler, more abstract proof of the Cayley-Hamilton theorem for matrices?

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 ...
1
vote
1answer
59 views

Roots of the characteristic polynomial of a symmetric matrix

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 ...
2
votes
0answers
187 views

Why matrix representation of convolution cannot explain the convolution theorem?

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 ...
2
votes
3answers
1k views

Insightful proofs for Sherman-Morrison Formula and Matrix Determinant Lemma

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 ...
9
votes
2answers
694 views

Algebraic proof of a trig matrix identity?

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 ...