Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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197
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Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\...
36
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1answer
1k views

Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
22
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1k views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
22
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0answers
794 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $$V= \begin{bmatrix} ...
18
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0answers
395 views

Can some proof that $\det(A) \ne 0$ be checked faster than matrix multiplication?

We can compute a determinant of an $n \times n$ matrix in $O(n^3)$ operations in several ways, for example by LU decomposition. It's also known (see, e.g., Wikipedia) that if we can multiply two $n \...
18
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0answers
393 views

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of $...
16
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0answers
263 views

Find the cardinality of a subset of $GL_n( \mathbb F_p)$

Let $m,n \in \mathbb N$. Let $\mathbb F_p$ denote the prime field of characteristic $p$. Consider the set $$ X_m = \{A \in GL_n( {\mathbb F_p}): A^m=1 \}$$ Compute the cardinality of $X_m$. Its ...
15
votes
1answer
390 views

The number of non-singular $n\times n$ matrices over $\mathbb{F}_2$ with exactly $k$ non-zero entries

Suppose $M_{n}^{k}$ is the number of non-singular $n\times n$ matrices over $\mathbb{F}_2$, that have exactly $k$ non-zero entries. Is there some sort of formula to calculate $M_n^k$? If $k < n$ ...
15
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0answers
1k views

Matrix generated by prime numbers

Let $p$ be the vector of dimension $n^2$ consisting of ordered prime numbers i.e. $p= [ 1 \ 2 \ 3 \ 5 \ 7 \ldots]^T$ and $A$ be the matrix of dimension $n\times{n}$ constructed with this vector by ...
13
votes
1answer
316 views

Upper bound for the widest matrix with no two subsets of columns with the same vector sum

Over at PPCG there is an ongoing contest going on to find the largest matrix without a certain property, called property $X$. The description is as follows (copied from the question). A circulant ...
12
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0answers
324 views

Bounding the minimum singular value of a block triangular matrix

Question: What is the sharpest known lower bound for the minimum singular value of the block triangular matrix $$M:=\begin{bmatrix} A & B \\ 0 & D \end{bmatrix}$$ in terms of the properties ...
12
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6k views

Simpson's Rule for Double Integrals

Simpson's Rule for double integrals: $$\int_a^b\int_c^df(x,y) \,dx \,dy$$ is given by $$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1} f(x_i,y_j) $$ where: $$W= \begin{pmatrix} 1&...
12
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0answers
468 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
12
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0answers
1k views

Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 &...
12
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0answers
887 views

Pfaffian properties

Given a $2n\times 2n$ real skew-symmetric matrix $A$, its Pfaffian $\mathrm{Pf}$ is defined as: $$ \mathrm{Pf}(A) = \frac1{2^n n!}\sum_{\sigma\in S_{2n}}\mathrm{sgn}(\sigma)\prod_{i=1}^n A_{\sigma(2i-...
11
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0answers
322 views

A conjecture on the Lyapunov equation

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e., all the eigenvalues of $A$ have strictly negative real part). Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix of ...
11
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0answers
216 views

Finding the ratio between two $8$-dimensional volumes

EDIT: At this point, geometric interpretations of conditions 2-4 would qualify as an answer. This can include symmetries of the region. I have a real $3 \times 3$ matrix $A$ with entries $a_{ij},$ ...
11
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0answers
249 views

Closeness the eigenvalues of a Matrix to a list of numbers

Let $A,B\in M_n(\mathbb R)$, and $\ell$ a list of $n$ numbers sorted in some order (say, decreasing). Let $\lambda_i(A)$ be the $i$th eigenvalue of $A$ with respect to the chosen order. Finally, let ...
10
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0answers
94 views

Minecraft water spreading initial arrangements

So as most minecraft players quickly learn, the best way to flood an area is to place water along a diagonal. There are of course other ways to flood the same area with the same number of buckets ...
10
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0answers
375 views

Flipping Summation of Kronecker Products

Question Suppose $\mathbf A$ is an $n\times n$ matrix, and that $\mathbf B_i$ is an $m\times m$ matrix, for all $i\in\{1,\dots, n\}$. Is it possible to find $n\times n$ matrices $\mathbf U$ and $\...
10
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0answers
223 views

How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
10
votes
2answers
387 views

What is known about the eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are the elements of the ...
10
votes
1answer
368 views

Determining sign(det(A)) for nearly-singular matrix A

Motivation: determining whether a point $p$ is above or below a plane $\pi$, which is defined by $d$ points, in a $d$-dimensional space, is equivalent to computing the sign of a determinant of a ...
10
votes
0answers
315 views

Uniqueness of an infinite system of linear ODEs

How to prove that $\dot{x}=ax,\space x(0)=1$ has a unique solution if $a,x$ are infinite dimensional matrices? More specifically, let $Q$ be a bounded infinitesimal generator, i.e. $Q=(q_{i,j})_{i,j\...
10
votes
0answers
468 views

Representation for Vandermonde's permanent

Permanent of a matrix A = $\|a_{i,j}\|_{i,j=1}^{n}$ is defined as $$ \mathrm{Perm}(A) = \sum\limits_{\sigma \in S_{n}} a_{1,\sigma_{1}},\ldots,a_{n,\sigma_{n}} $$ Is there some representation for ...
9
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0answers
148 views

Reference request: arranging distinct numbers into a full rank matrix

I thought of the following problem: Let $n\ge 2$. Suppose you have $n^2$ distinct numbers in some field. Is it necessarily possible to arrange the numbers into an $n\times n$ matrix of full ...
9
votes
1answer
195 views

Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

The set of Schur stable matrices is \begin{align*} \mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\}, \end{align*} where $\rho(\cdot)$ denotes the spectral radius of a matrix and the set of ...
9
votes
0answers
109 views

What is $\ \overline{\bigcup_{p≥ 1}\ \{A\in M_n(\mathbb C), \ A^p = I_n\}} \ $?

Let $\Gamma_p = \{A\in M_n(\mathbb C), A^p = I_n\}$ and let $\Gamma = \bigcup_{p≥ 1}\ \Gamma_p$. What is the closure of $\Gamma$ ? (This is from an oral exam). Let $B \in M_n(\mathbb C)$ such that ...
9
votes
0answers
148 views

Matrix exponential, containing a thermal state

Define an infinite matrix $$ M = \begin{bmatrix} 0 & -1 & 0 & 0 & \cdots \\ 1 & 0 & -2 & 0 & \cdots \\ 0 & 2 & 0 & -3 & \cdots \\ 0 & 0 & 3 &...
9
votes
1answer
419 views

Generalisation of prime numbers to matrices?

Is it possible to generalise prime numbers to matrices? I'm trying to solve a Rubix cube in the minimum number of steps and I think this would be useful. I think it's possible to represent Rubix cube ...
9
votes
0answers
324 views

Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?

I realize that asking this question is like presenting to a patent attorney a wheel-less skateboard while asking to patent a hoverboard. Anyways. Lagarias version of the Riemann hypothesis sets a ...
9
votes
1answer
171 views

Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ \...
9
votes
1answer
289 views

Expressing the values of a matrix at pow N

I have a square matrix (that comes from a Markov Chain) that looks like that: $$Q = \begin{bmatrix} 0 & 1& 0 & 0 & .. & 0 & 0\\ 0 & a & 1-a & 0 & .. & 0 ...
9
votes
1answer
695 views

Cones of positive semidefinite matrices generated by matrices of rank $1$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive ...
8
votes
0answers
227 views

(How) can one visualise the derivative of the function $A \mapsto A^{-1}$, where $A$ is a matrix?

Our definition of differentiable is: Let $V$ and $W$ be Banachspaces, $V$ finite dimensional, and $G \subset V$ an open subset. We call a function $f: G \to W$ differentiable in $p \in G$ if there ...
8
votes
0answers
77 views

Determinant of a symmetric matrix with entries on diagonals

I am interested in the calculation of the determinant of the $N\times N$ symmetric matrix \begin{equation*} \mathbf B = \left(\begin{array}{*{20}c} 2 & & -1& &-1& &\\ & 2 &...
8
votes
0answers
239 views

When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is related to the question How many connected components for the intersection $S \cap GL_n(\mathbb R)$ where $S \subset M_n(\mathbb R)$ is a linear subspace? I asked. There is a nice example in ...
8
votes
0answers
311 views

Polynomial approximation to formal power series matrix

I noticed that starting with a $2 {\times} 2$ matrix $M$ with a handful of polynomial entries in two variables such that $\det M$ is invertible in $\mathbb C [x] [[y]]$ I can add infinitely many terms ...
8
votes
0answers
147 views

The practical usage of Arnold Matrix Trace Theorem

I would like to ask about the Arnold's Matrix Trace theorem: $$\textrm{tr}\big(A^{p^k}\big)\equiv\textrm{tr}\big(A^{p^{k-1}}\big)\ (\!\!\bmod {p^k}).$$ This theorem looks fantastic to me. But is ...
8
votes
1answer
98 views

$\left[\begin{array}{cc} P & A^T\\ A & 0\end{array}\right]$ is non-singular if and only if $\mathcal N(P) \cap \mathcal N(A)=\{0\}?$

Suppose $P\succeq 0$ and $A$ is of full row rank. I want to show that $\left[\begin{array}{cc} P & A^T\\ A & 0\end{array}\right]$ is nonsingular if and only if $\mathcal N(P) \cap \mathcal N(...
8
votes
0answers
2k views

Upper bound on Frobenius norm of inverse of positive definite, symmetric matrix

Let $\Sigma$ be a symmetric, positive definite $n \times n$ matrix. I want an upper bound on the Frobenius norm of $\Sigma^{-1}$ that does not involve calculating the determinant of $\Sigma$. The ...
8
votes
0answers
2k views

Sylvester's criterion for negative semidefinite matrices

Is there a Sylvester's criterion for negative semidefinite matrices? I suspect such a criterion to be: All principal minors with odd dimension are non-positive. All principal minors with even ...
8
votes
0answers
3k views

Is a symmetric positive definite matrix always diagonally dominant?

A Hermitian diagonally dominant matrix $A$ with real non-negative diagonal entries is positive semidefinite. Is it possible to have a Hermitian matrix be positive semidefinite/definite and not be ...
8
votes
0answers
4k views

Matrix multiplication of columns times rows instead of rows times columns

In ordinary matrix multiplication $AB$ where we multiply each column $b_{i}$ by $A$, each resulting column of $AB$ can be viewed as a linear combination of $A$. If however if we decided to multiply ...
8
votes
0answers
5k views

What do you call the product of a matrix's diagonal elements?

The trace of $A$, an $N\times N$ matrix, is $\displaystyle\sum_{i=1}^N A_{ii}$. What do you call $\displaystyle\prod_{i=1}^N A_{ii}$?
8
votes
1answer
754 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
7
votes
1answer
91 views

Degree of the kernel of a module map $R^n\rightarrow R^n$ for an Euclidean domain $R$

Let $R$ be an Euclidean domain with the degree function $d$. Let $A\in R^{n\times n}$ be an $n\times n$-matrix with entries in $R$ such that det$(A)=0$. As a module map $A:R^n\rightarrow R^n$, there ...
7
votes
0answers
218 views

If nilpotent matrix $A$ and $AB−BA$ commute, show that $AB$ is nilpotent.

Let $A$ and $B$ be $n×n$ complex matrices. If $A$ is a nilpotent matrix, and $A$ commute with $AB−BA$, show that $AB$ is nilpotent. Equivalently, the question can be expressed as following ...
7
votes
0answers
150 views

Meaning cone,ray, fan for polytopes

I'm trying to understand some mathematical operations and definitions for my project. Could you explain the meanings? $P_b ={x∈R^d : Ax≤b}$ is a polytope. Let's have a 10x3 matrix $Ax≤b$ $$ \begin{...
7
votes
0answers
167 views

Theorem 2.20 in Friedberg's Linear Algebra

Theorem 2.20 Let $V$ and $W$ be finite-dimensional vector spaces over $F$ of dimensions $n$ and $m$, respectively, and let $\beta$ and $\gamma$ be ordered bases for $V$ and $W$, respectively. Then the ...