# Questions tagged [matrices]

Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.

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### What is the correct approach to calculate the determinant of a matrix? [closed]

I am a neophyte undergraduate student reviewing multivariate analysis. I am learning from Mathematical Tools for Applied Multivariate Analysis (J.D. Caroll, P.E. Green). I covered the co-factor ...
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### Continuity of eigenvector of zero eigenvalue

Wonder whether anyone has an idea on showing the following or to point out that it is not true: Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
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### What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?

Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation \begin{align*} & X = A X A^T + \operatorname{Id} \tag{1} \...
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### Norm of a matrix function of a vector $x$

I have a matrix $$A(x) = \frac{-1}{(1+\|x\|_2^2)^{\frac{3}{2}}}xx^T + \frac{1}{(1+\|x\|_2^2)^{\frac{1}{2}}}I$$ I see that $$\|A(x)\|_2 \le 1 \ \forall x$$ But is $\|A(x)\| \le 1$ in general $\forall x$...
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### “Smallest” non-zero linear combination of vectors to obtain a non-negative vector

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix} where $x_{j} \geq 0$. Suppose that ...
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### Quick calculation of a symmetric product with two indices

Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
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### What is the impact of individual estimate on each other in matrix inversion?

I am looking to understand the impact of each estimate on each other in matrix inversion. Lets say I have a vector $A = \left[a_1, a_2 \right]^T$ of size $2 \times 1$ and $a_1$ and $a_2$ are related ...
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### Matrix-order derivatives (differentiating a function a matrix number of times)

I have been exploring methods of generalizing the order of derivatives to a broader range of inputs (such as real numbers, complex, and now matrices). We are very well familiar with integer-order ...
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### Interpolation between two matrices so that $L^p$ norm is controlled

Assume to have two square matrices $A$ and $B$ acting on $\mathbb{R}^n$ such that all their entries are in the interval $[0,1]$ and such that $||Ax||_1 = ||x||_1$ and $||Bx||_\infty \leq ||x||_\infty$....
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### What is the mathematician's definition of the determinant? [closed]

I am trying really hard to find a good definition of the determinant. I have looked virtually every single resource online and everybody gives a different answer: sum of cofactors or minors https://...
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### Efficiently solve the Sylvester equation $AX+XA = C$ where $X$ is skew-symmetric

Is there a way (more efficient than the standard vectorization) to solve the following Sylvester equation in the skew-symmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive ...
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### Two questions about three circulant matrices

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$ $$2AA^T+BB^T+CC^T=(4n+4)I-4J$$ where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
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### Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?

I have a matrix that is “kind of symmetric.” Specifically, it is an $n \times n$ real matrix such that the entries $a_{ij}=1/a_{ji}$ whenever $j \ne i$. I want to investigate the properties of this ...
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### Conditions on symmetric $3 \times 3$ matrices to satisfy the convex equality for cofactor and determinant

Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such ...
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### On a matrix equation with Kronecker product

Is there any work on the matrix equation in unknowns $X, Y \in {\Bbb C}^{n \times n}$ $$(X \otimes Y + Y \otimes X) \operatorname{vec}(A)=0$$ where $\otimes$ is the Kronecker product? Or, in general, ...
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### Expressing an invertible sparse matrix as a product of few elementary matrices

Let $M$ be an $n \times n$ matrix with integer entries. Suppose that $M$ is invertible (over the integers) and that $M$ has at most $An$ nonzero entries, each of which is less than $B$ in absolute ...
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### Symmetric indefinite matrix of fixed rank — manifold structure?

I have been studying symmetric indefinite matrices of fixed rank, which have been rather useful for a particular application. I wonder if there is a way to parameterise these by a smooth manifold, e.g....
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### Change of the smallest positive eigenvalue after a rank-one update

Given natural numbers $n,r,R\in\mathbb{N}$ with $r,R\le n$, let $A\in\mathbb{R}^{n\times r}$ and $B\in \mathbb{R}^{n\times R}$ be two matrices with full column rank and let $c\in\mathbb{R}^n$. Denote ...
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### Non-vanishing principal minors up to swapping columns

An undergraduate student asked me the following seemingly easy question. After a few days of thinking, I still couldn't come up with an answer, nor could I find one online. Maybe folks here could help?...
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### Are these two methods for constructing Hadamard matrices known?

These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers: Context: Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this ...
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### Pre-positive definite functions?

A function $f(x,y)$ is positive definite if matrices $( f(x_i, x_j) )_{i, j \in F}$ are positive definite for all finite index sets $F$. This is frequently hard or impossible to check given some ...
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### Existence of cyclic subspace decompositions for pairs of commuting matrices

Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute. For $v\in V$, ...
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### Interpretate a condition on Signed graph

In my research I came up with a condition on signed complete graph as follows: Consider adjacency matrix of signed complete graph, i.e. symmetric, diagonal-free matrix with elements $A_{ij}=-1$ or $+1$...
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### Inductive Cholesky decomposition to prove that a function is positive definite over the natural numbers?

I am trying to prove that the function: $$k(a,b):=\frac{1}{\operatorname{rad}\left ( \frac{ab}{\gcd(a,b)^2} \right )}$$ is a positive definite function over the natural numbers. What has sometimes ...
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### Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution. What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...