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Maybe it is not really suitable to undergrads (unless they are really problem solving-oriented), but there is a nice proof that $$\prod_{1\leq i \lt j\leq n} \frac{x_j-x_i}{j-i}$$ is integer for all i …

This is essentially the same as Tobias Hagge's answer and Jonny Evans's comment, but I thought that writing it up in this way would make things clearer. Think about the product $$ \begin{bmatrix} a & …

The key word under which you will find this result in modern books is "Schur complement". Here is a self-contained proof. Assume $I$ and $J$ are $(1,2,\dots,k)$ for some $k$ without loss of generality …

You have determined $A^*A$, or, alternatively, you know $A$ up to pre-multiplication by a unitary matrix $U$. So you know the $R$ factor of its QR factorization, and the factors $\Sigma$ and $V$ of it …

Disclaimer 1: Treating these topics properly would require a quick course in numerical analysis. Disclaimer 2: If you are using any sane computer system, it's already going to have a library function …

Implementing the QR factorization with Householder rotations is cheaper ($2n^2m$ vs $3n^2m$ for a $m\times n$ matrix), and equally accurate in practice. See Section 19.6 of Higham's Accuracy and Stabi …

If I understand correctly, you will find the answer in a good exposition of the Banach-Tarski paradox. Finding two rotation matrices in $\mathbb{R}^3$ that generate the free group in two generators is …

Yes. Every matrix can be written as the sum of a symmetric plus an antisymmetric one: $A = \frac{A+A^T}{2}+\frac{A-A^T}{2}$. Now change basis such that the symmetric part is diagonal.

Yes! Most methods to compute exponentials of large sparse matrices are based on computing directly $\exp(A)b$ for a given vector $b$ rather than the full matrix $\exp(A)$. Just take $b$ as a vector of …

It is a principal pivot transform, also known as sweep operator or gyration. You can check the linked review paper.

it is called "diagonal congruence" here. This makes sense, at least when $D$ is real, since it is a congruence. "Conjugate" sounds more like $D^{-1}AD$ or $\overline{A}$ to me.

These matrix equations are called Lyapunov equations and are extensively studied in control theory. For instance, if $A$ is Hurwitz (all eigenvalues in the left half-plane), then the unique symmetric …

A simpler, more direct proof that requires no SVD: let $Y_j$ be the $j$th column of $Y$ and $Z_j$ that of $Z=XY$. Then, $$\|Z\|_F^2 = \sum_j \|Z_j\|_2^2 = \sum_j \|XY_j\|_2^2 \leq \sum_j \|X\|_2^2\|Y_ …

The sum of the elements of a matrix $M$ is $e^T M e$, where $e$ is the vector of all ones. So, instead of computing the inverse, you should solve the system $Ax=e$ and then compute $e^Tx$. This might …

There are 5 inequivalent Hadamard matrices of order 16; if I understand correctly that's a counterexample.