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…

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 $A$ 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 $A,B\geq 0$, 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 $A,B\in M_n$,

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

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

Let $\mathbb{K}=\mathbb{R}$ 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…

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…

Let ${\cal H}_n$ be the cone of positive-semidefinite Hermitian matrices, and let $A\circ B$ denote the Hadamard (entrywise) product. Chollet's conjecture. For $A,B\in{\cal H}_n$,…

Marcus's block permanent conjecture. Then

Let $A$ be partitioned into square blocks as … where $A$ is positive-semidefinite Hermitian. Marcus–Newman conjecture. For this partition, … The conjecture was solved by the coroll…

Hadamard-type inequality for hyperbolic polynomials. The claim extends the diagonal-versus-eigenvalue inequality from elementary symmetric polynomials to homogeneous, real, symmetr…