55 problems

Let $M_{m,n}(\mathbb{C})$ denote the space of complex $m\times n$ matrices. For a matrix, write $\|\cdot\|_F$ for the Frobenius norm, $\|\cdot\|_2$ for the spectral norm, and…

Let $\mathbf{X}_1,\dots,\mathbf{X}_n$ be i.i.d. matrices with common mean matrix $\mathbb{E}\mathbf{X}_1=\mathbf{M}$ and $p$th central moment matrix … For any…

Let $M$ be any $n\times n$ symmetric matrix with entries satisfying $0\leq M_{ij}\leq1$, and let $\mathbf{1}$ be the $n$-dimensional all-ones vector. Matrix power inequality conjec…

Let $n$ be a positive integer, $A_1,\dots, A_n$ be symmetric positive semidefinite matrices, and let $\lVert\,\cdot\,\rVert$ be the spectral norm. For $m\leq n$, compare the average…

Let $M(n,\mathbb{R})$ be the space of real $n\times n$ matrices. Let $B,B_2,\ldots,B_m\in M(n,\mathbb{R})$, with $B_\alpha^*$ denoting transpose and with the Frobenius inner produc…

Let $M(n,\mathbb{R})$ be the space of real $n\times n$ matrices. For $X\in M(n,\mathbb{R})$ with Frobenius norm $\|X\|=1$, let $T_X(Y)=[X^*,[X,Y]]$ and let $\lambda(T_X)$ denote it…

Conjectured tight bounds. The tight bounds for cases (i), (ii), and (iv) are respectively

Let $A$ be a positive semidefinite Hermitian matrix, let $\lambda\vdash n$ be a partition, let $\chi^{\lambda}$ be the irreducible character of $\mathfrak{S}_n$ indexed by…

Let $X$ and $Y$ be two $n\times n$ matrices, let $[X,Y]=XY-YX$ be their commutator, and let $||\cdot||$ denote the Hilbert–Schmidt norm. Böttcher–Wenzel conjecture. … The conjectur…

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Lu–Wenzel eigenvalue conje…

Let $\mathbb{K}=\mathbb{R}$ and let $B_1,\ldots,B_m\in M(n,\mathbb{K})$, where $M(n,\mathbb{K})$ is the space of $n\times n$ matrices over $\mathbb{K}$. Let $[A,B]=AB-BA$ and let…

Lin's conjecture. There exists a unitary matrix $U\in M_n$ such that

Let $M$ be a matrix with non-negative real entries, let $t$ be an odd positive integer, and let $(x_1,y_1),\ldots,(x_t,y_t)$ be distinct entries of $M$ satisfying $M_{x_i y_i}=0$ f…

Weak Loewner spectral-dominance conjecture. The matrices $L$ and $D$ satisfy

Le's conjecture. Then $\operatorname{diag}(A)\in\Gamma(P)$ and

Let $Z$ be an $n\times n$ positive semidefinite similarity matrix as above, and let $\mathbf{1}$ denote the $n$-dimensional all-ones vector. Equivalent bounds conjecture. For $q>1$…

Let $Z$ be an $n\times n$ positive semidefinite similarity matrix with $n>0$, symmetric entries in $[0,1]$, and diagonal entries equal to $1$. On the uniform distribution, write LC…

Let $A,B\geq 0$, and let $p,q,r$ be real parameters satisfying $0<p\leq q$ and $r\geq 0$. Here $\succ_{\log}$ denotes log-majorization. Reverse log-majorization conjecture. One has…

Lee's conjecture. For all $A,B\in M_n$,

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$, and let $B,B_2,\ldots,B_m\in M(n,\mathbb{K})$. Let $B^*$ denote the conjugate transpose and let $[A,B]=AB-BA$. Assume … for…

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Ge–Li–Lu–Zhou conjecture.…

Let $\mathbb{K}=\mathbb{R}$ and let $B,B_2,\ldots,B_m\in M(n,\mathbb{K})$. Let $B^*$ denote the conjugate transpose, let $\bot$ denote orthogonality for the Frobenius inner produc…

Let $X$ and $Y$ be arbitrary real Toeplitz matrices, let $\langle X,Y\rangle$ denote their Frobenius inner product, and let $\|\cdot\|$ denote the Frobenius norm. Lu–Wenzel conject…

Bourin–Lee's multivariable conjecture. The unitary matrices can be chosen so that

Let $A\in{\cal H}_n$, and let $A(i|j)$ denote the matrix obtained by deleting row $i$ and column $j$. Pate's inequality challenge. … The source presents this as a consequence of th…