Hello all, This is a question that might or might not be related to my previous one. Imagine you have two matrices: Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L …

Given an invertible matrix $A \in \mathbb{R}^{n \times n}$, and index set $\langle n\rangle = \{ 1, \dots, n \}$, and the submatrix $A(\alpha)$ with the columns and rows of $A$ wit …

Define $\rho(A)$ to be the spectral radius of a square matrix $A$. Let $S$ and $T$ be two non-negative square matrices and $h$ a real number such that $\rho(S+T) < h$. Show that …

I am interested in estimating how large determinants of matrices tend to be 'on average' given the following model: suppose we form $n \times n$ matrices $M$ such that all of the e …

Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$. Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that …

Hi Suppose that $A$ is a symmetric positive semi-definite matrix and $B$ is a square matrix (not necessarily symmetric) with the real parts of all the eigenvalues positive. Will a …

Is the DFT matrix the unique* unitary matrix with all entries of same magnitude? (*up to some trivial transformations)

Hello, I am concerned with a rectangular m by n matrix M, with entries picked as Gaussian random variables (with some fixed mean and variance). How large does m need to be so tha …

What can one say about a matrix for which the inverse of each leading principal minor is equal to the leading principal minor of its inverse (of the same dimensions, of course)? Fo …

Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D …

If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B. Given then all these 2x2 det …

A famous problem in linear algebra is that "A $n\times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$." I know one proof using Smith Normal form. Howeve …

It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way: $|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$. Suppose we define a dif …

Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$. One of my friend asked me …

Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$, $$f(x,y) = e^{\imath\pi x g(y)}$$ where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\le …