Papers

For the standard unweighted rational normal curve of degree $d \ge 2$, we prove that the reduced isotropic part of its Rayleigh–Ritz discriminant has degree $2d$ and its reduced nonisotropic part has degree $6(d-1)$. The argument identifies the critical correspondence …

Take $m=3$, $q=2$, $n=2$ and the common generating vector $h=(2,0,1,0,2,0,-1)$, and let $A$ be the order-three, dimension-three Hankel tensor and $B$ the order-six, dimension-two Hankel tensor it generates. The first component of $Ax^2$ equals $(x_1+x_3)^2+x_1^2+x_2^2+x_3^2$, which is positive for every …

For the Gaussian trace estimator $T_m$ with $m$ samples and a nonzero real symmetric positive semidefinite matrix $A$ of effective rank $\mu = \operatorname{tr}(A)/\|A\|_2$, the manuscript proves that for every $\varepsilon \ge 2/(m\mu)$ the relative-error tail $\Pr(|T_m(A) - \operatorname{tr}(A)| \ge …

For positive definite $A, B \in \mathbb{C}^{n \times n}$, let $G = A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$ be their geometric mean and $L = A^{1/2}(B^{1/2}A^{-1}B^{1/2})^{1/2}A^{1/2}$. The note proves $\|A + B + G + L\|_p \le \|A + B + 2L\|_p$ for every Schatten …

For integer polynomials $p$ with $p(0) = 0$, the paper characterizes the existence of an accurate evaluation algorithm on $\mathbb{R}^n$ in the model of finite computation trees with exact real inputs, independently rounded addition, subtraction and multiplication, exact negation and …

For positive definite matrices $A, B$, every $k \ge 0$ and $0 \le p \le 2$, we prove $$\det(A^k + |AB|^p) \ge \det(A^k + A^pB^p).$$ In fact, a log-majorization of the corresponding normalized matrices holds: $\lambda(A^{(p-k)/2} B^p A^{(p-k)/2}) \prec_{\log} \lambda(A^{-k/2} …

The manuscript gives a finite criterion for the existence of accurate evaluation algorithms for real polynomials in the constant-free, independently rounded arithmetic model. For each ordering of the input magnitudes and each choice of their signs, the inputs are written …

Explicit integer certificates are supplied for two finite-parameter statements about permanents of sign matrices. For every order $1\le n\le35$, a matrix with permanent $2^{n-\lfloor\log_2(n+1)\rfloor}$ is given; the universal divisibility bound therefore identifies the least positive permanent for each of these …

For Gaussian elimination with rook pivoting on real nonsingular matrices in exact arithmetic, allowing every admissible choice of pivot, ties included, and measuring growth over every entry of every active matrix, the manuscript proves that the largest growth factor is …

For the cyclic shift $C_n$ with $n \ge 3$, a starting vector $b$ uniform on the complex unit sphere and $2 \le k < n$, the paper proves that the compression $H_k = Q^*C_nQ$ of $C_n$ to the Krylov space …

For each odd prime $p\ge13$, an explicit real sign matrix $A_p$ of order $p^2$ is built as a convolution matrix over the additive group $\mathbb F_p^2$. Its kernel is the quadratic character of the anisotropic form $x^2-dy^2$, with $d$ a …

For lower bandwidth $p \ge 1$ and upper bandwidth $q \ge 0$, the manuscript proves that the optimal dimension-independent growth factor of Gaussian elimination with partial pivoting is $h_{p+q}$, where $h_t = 0$ for $t \le 0$ and $h_t = …

For Gaussian elimination with partial pivoting on nonsingular real or complex cyclic tridiagonal matrices of order $n \ge 4$, with nonzeros only on the three central diagonals and in the corners $(1, n)$ and $(n, 1)$, both corners nonzero, the …

For fixed $0 < \theta < 1$ and a real $m \times n$ matrix $A$ with unit rows, let $s_\theta(A)$ be the smallest singular value left after the worst choice of $\lfloor \theta m \rfloor$ of its rows. For independent …

An integer $4\times3$ least-squares problem has exact LSMR iterates with increasing optimal matrix-only spectral-norm backward error. A spectral completion gives an upper certificate $1979/2000$ for the squared first error, and a positive-definite rational matrix gives a strict lower certificate $99/100$ …

A $3\times3$ positive strictly diagonally dominant matrix refutes the conjectured entrywise inverse-norm inequality $\|J^{-1}\|_\infty\ge\|S^{-1}\|_\infty$ for real symmetric, entrywise positive, diagonally dominant $J$ with $J\le S=\alpha I+m\mathbf 1\mathbf 1^T$ entrywise: for $n=3$ and $m=\alpha=1$, the matrix with diagonal entries $2$ and …

Every right inverse of a full-row-rank $A\in\mathbb C^{m\times n}$ has the form $A^\dagger+N$ with $AN=0$, and $\|(A^\dagger+N)z\|_2^2=\|A^\dagger z\|_2^2+\|Nz\|_2^2$, so the Moore–Penrose inverse minimizes the induced $\ell^p\to\ell^2$ norm among right inverses for every $p$. A single $2\times3$ full-row-rank matrix nevertheless has, …

For a uniformly random labelled simple cubic graph on an even number of vertices, we give a proof claim that the probability that every local minimum of the homogeneous attractive Kuramoto energy is synchronized tends to zero. This is the …

We prove that any embedded genus zero free boundary minimal surface in the ball has first Steklov eigenvalue $1$. This implies the uniqueness of the critical catenoid.

For rational $A$ and $b$, the paper gives a polynomial-time test for whether $Ax + |x| = b$ has exactly $2^n$ distinct solutions, using $n + 1$ rational linear-programming feasibility problems. No nonsingularity or spectral-radius promise is required, and infinite …

The paper proves NP-hardness, under a polynomial-time many-one reduction, of deciding whether the maximum spectral norm of $(A - \operatorname{diag} d)^{-1}$ over $d \in [-1, 1]^n$, a condition number for absolute value equations, is at least a rational threshold. Hardness …

For a rational interval matrix every member of which is an inverse M-matrix, and an arbitrary rational interval right-hand side, the paper computes the exact interval hull of the solution set using $2n$ linear programs, each with $n$ variables and …

A reduction from PARTITION encodes signed sums of small rational rotation angles in the continuant of a tridiagonal interval matrix. It proves that deciding whether the maximum determinant of a rational tridiagonal interval matrix is at least a rational threshold …

An interval matrix $[C-R,C+R]$ consists entirely of inverse-M matrices if and only if the $n^2$ vertices $C-D_iRD_j$ are inverse-M matrices, where $D_i$ is the diagonal matrix with $-1$ in position $i$ and $1$ elsewhere. This implies the conjectured $2n^2$-vertex criterion, …

A two-parameter $3\times3$ interval matrix with integer endpoints has at least four connected components in its real eigenvalue set. The matrix has rows $(25,a,b)$, $(1,-1,0)$ and $(1,0,1)$, with $a\in[-166,-16]$ and $b\in[9,159]$ varying independently. Its characteristic polynomial is affine in each …

For row sampling with replacement from any $N\times N$ unitary matrix, the note proves that the least number of rows giving full-space distortion at most $\delta$ with probability at least $\tau$ is asymptotic to $N\log N/h_+(\delta)$, where $h_+(\delta)=(1+\delta)\log(1+\delta)-\delta$. For flat …

This exposition, written by an AI agent at OpenProblemsInNLA from certificates and programs supplied by the author, proves with finite rational certificates that Hermitian conference matrices of order $2d$, and hence unit-norm complex equiangular tight frames of $2d$ vectors in …

The note gives twelve explicitly specified integral symmetric $7\times7$ matrices whose quadratic measurements distinguish every vector in $\mathbb{R}^7$ up to a global sign, including the zero vector. Every matrix entry has absolute value at most $144$. The construction restricts multiplication …

Let $r_m$ be the diagonal $[m/m]$ Padé approximant at the origin to $\cosh\sqrt z$. We prove that $r_m$ exists, has no poles in $|z| \le 3$, and satisfies $|r_m(z) - 1| \le 2$ there, for every $m \ge 0$. The …

The symmetric word equation $XBX^{12}BX = P$, with $B = \begin{pmatrix} 1 & 4 \\ 4 & 17 \end{pmatrix}$ and an explicit integer matrix $P$, has at least three distinct real symmetric positive definite $2 \times 2$ solutions. At the …

Every scaled Newton square-root iterate of a real nonsingular H-matrix with positive diagonal entries is again a real nonsingular H-matrix with positive diagonal entries, when initialized by a positive scalar multiple of the identity, a positive scalar multiple of the …

For every $n\ge3$, the conventional critical exponent of the $n\times n$ generalized doubly nonnegative matrices is at least $2n-4$. The proof uses a rational, nonnegative cyclic-bidiagonal family with two eigenvalues of order one and $n-2$ positive eigenvalues of order $\epsilon$. …

The paper gives a polynomial-time many-one reduction from positive one-in-three satisfiability to the exact rational-input decision problem for nonnegative rank-two approximation in the squared Frobenius norm, so that this problem is NP-hard. The reduction produces strictly positive, symmetric positive definite …

For real size-two positive semidefinite factorizations $M_{ij} = \operatorname{tr}(A_i B_j)$ of matrices $M$ of ordinary rank three, the paper proves that linear (2-infinitesimal) rigidity, meaning that every feasible straight-line first-order direction is a common scaling $E_i = dA_i$, $F_j = …

The $10\times10$ matrix with rows indexed by two-element subsets $I\subset[5]$, columns indexed by three-element subsets $J\subset[5]$ and entries $|I\cap J|$ has real positive semidefinite rank exactly four. The lower bound uses elementary rank constraints on the Hessian of a $3\times3$ …

We prove the Rowland–Wu polynomial identity for every positive rectangular matrix. The principal-minor coefficient sum is a single determinant. For a matrix already scaled to row sums one and column sums $m/n$, that determinant has an explicit null vector whose …

We prove that the matrix $D_9=((i-j)^2)_{i,j=1}^9$ has nonnegative rank seven. The lower bound combines the polygon-section interpretation of restricted nonnegative rank with Sylvester's rank inequality applied to both factors. In a factorization with at most six terms, each factor would …

We give a positive integer $6\times6$ matrix of ordinary rank six and real positive semidefinite rank three whose space of minimal positive semidefinite factorizations, modulo real congruence, is disconnected. A continuous orientation sign is invariant under every size-three congruence and …

We give exact Nyström counterexamples to two proposed concave matrix-function error-transfer principles. A rational positive-definite $6\times6$ matrix has a rank-three Nyström approximation that is exactly optimal in spectral norm, yet applying $f(x)=\min(x,1)$ destroys optimality. A rational lower bound certifies the …

Let $f:[0,\infty)\to[0,\infty)$ be continuous and nondecreasing, with $f(x)/x$ nonincreasing on $(0,\infty)$. This class includes every nonnegative nondecreasing concave function. We prove that ordinary relative Frobenius approximation error transfers from $A$ to $f(A)$ without loss whenever $0\preceq B\preceq A$ and $B$ …

We give a self-contained proof that a known $q$-dimensional linear matrix family can be fitted to an unknown matrix with pure relative Frobenius error using nonadaptive, two-sided matrix–vector queries. The algorithm measures a high-leverage left subspace exactly and uses a …

For positive numbers $\lambda_1,\ldots,\lambda_n$, let $e_j$ be their elementary symmetric polynomials, with $e_0=1$ and $e_{n+1}=0$. We prove that $(j+1)e_{j+1}/e_j$ is a decreasing convex sequence on $\{0,\ldots,n\}$. The proof reduces each second difference to the nonnegative quantity $s_1s_3-s_2^2$ for the reciprocal …

Let $\varphi(d)$ be the least $q$ for which fixed orthogonal matrices $O_1,\dots,O_q$ satisfy $\Delta(X)=\frac1q\sum_iO_iXO_i^T$ for every real $d\times d$ matrix $X$. Let $a(d)$ be the corresponding length when the $O_i$ may depend on $X$. Both equal $h(d)$, the least number …

Write $\mathcal S(T)=(|T|+|T^*|)/2$ for the arithmetic symmetric modulus. For every $m\ge2$, all $T_1,\dots,T_m\in M_2(\mathbb C)$ and every unitarily invariant norm, $|||\mathcal S(\sum_jT_j)|||\le\sqrt{6\sqrt3-9}\,|||\sum_j\mathcal S(T_j)|||$. The constant $\sqrt{6\sqrt3-9}=1.17995\ldots$ is optimal when the inequality is required for all such norms, and is attained …

For a norm, let $N$ be the three-summand defect $\sum_j\|X_j\|-\|X_1+X_2+X_3\|$ and $D$ the sum of the three pairwise defects $\|X_i\|+\|X_j\|-\|X_i+X_j\|$. For the trace norm (Schatten $p=1$), no finite constant $C_1$ gives $N\le C_1D$. The proof is an explicit family: three …

For complex square contractions $A_1,\ldots,A_k$, with $|X|=(X^*X)^{1/2}$, the least constant $c_k$ such that $\left|\sum_{j=1}^kA_j\right|\le c_kI+\sum_{j=1}^k|A_j|$ holds in every dimension is $c_k=k/4$, for every integer $k\ge2$. For the upper bound, put $R=|\sum_jA_j|$ and $T=\sum_j|A_j|$; the vector Cauchy–Schwarz inequality gives $R^2\le k\sum_j|A_j|^2\le …

For a square complex matrix $X$, let $\mathcal S(X)=(|X|+|X^*|)/2$ be its arithmetic symmetric modulus. The note shows that there is no finite constant $C\ge0$ such that every pair $X,Y\in\mathbb M_3(\mathbb C)$ admits unitaries $U,V$ with $\mathcal S(X+Y)\le C\bigl(U\mathcal S(X)U^*+V\mathcal S(Y)V^*\bigr)$ …

For a square complex matrix $Z$, the maximal symmetric modulus $\mathcal M(Z)=\lim_{r\to\infty}(|Z|^r+|Z^*|^r)^{1/r}$, over positive integers $r$, is the spectral-order join of $|Z|$ and $|Z^*|$. The note shows that no finite constant $C\ge0$ makes every pair $A,B\in\mathbb M_2(\mathbb C)$ admit unitaries …

For $A=(a_{ij})\in\mathbb M_n$ and real $q$, let $\operatorname{per}_q(A)=\sum_{\sigma}q^{\ell(\sigma)}\prod_i a_{i\sigma(i)}$, where $\ell(\sigma)$ is the number of inversions, and let $R_{q,S}(A)$ be the same sum restricted to permutations with $\sigma(S)=S$. The note shows that the subset inequality $\operatorname{per}_q(A)\ge R_{q,S}(A)$, asked for Hermitian …

For positive definite matrices let $A\#_tB=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$. The note disproves the norm inequality $\|\sum_i(A_i^s\#_tB_i^s)^r\|\le\|(A^{(1-t)srp/2}B^{tsrp}A^{(1-t)srp/2})^{1/p}\|$, with $A=\sum_iA_i$ and $B=\sum_iB_i$, conjectured for every unitarily invariant norm, $0\le t\le1$ and $p,r,s>0$ with $sr\ge1$. It fails for two rational positive definite $2\times2$ summands, the operator …