Jan 11, 2015 · there is a similar thread here Coordinate-free proof of $\operatorname {Tr} (AB)=\operatorname {Tr} (BA)$?, but I'm only looking for a simple linear algebra proof.

Feb 28, 2018 · How many 4 4 -digit palindromes are divisible by 3 3? I'm trying to figure this one out. I know that if a number is divisible by 3 3, then the sum of its digits is divisible by 3 3. All I …

Aug 29, 2013 · Given two square matrices A, B A, B with same dimension, what conditions will lead to this result? Or what result will this condition lead to? I thought this is a quite simple …

Can there be said anything about det(A + B) det (A + B)? If A/B A / B are symmetric (or maybe even of the form λI λ I) - can then things be said?

Given two square matrices A and B, how do you show that det (AB) = det (A) det (B) where det (⋅) is the determinant of the matrix?

I know that there are three important results when taking the Determinants of Block matrices $$\\begin{align}\\det \\begin{bmatrix} A & B \\\\ 0 & D \\end ...

The commutator [X, Y] of two matrices is defined by the equation $$\begin {align} [X, Y] = XY − YX. \end {align}$$ Two anti-commuting matrices A and B satisfy $$\begin {align} A^2=I \qu...

This question does a great job of illustrating the problem with thinking about these things in terms of coordinates. The thing that is positive-definite is not a matrix M M but the quadratic form x ↦ …

Jun 6, 2015 · We can define trace if A =∑i ei, Aei A = ∑ i e i, A e i where ei e i 's are standard column vectors, and x, y =xty x, y = x t y for suitable column vectors x, y x, y. With this set up, I …

As a remark, it is actually legitimate to assume that A A and B B are simultaneously diagonalisable (surprise, surprise!), so the proposition is trivial. But obviously, the reason why …