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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Probability for an $n\times n$ matrix to have only real eigenvalues

Let $A$ be an $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has only real eigenvalues? The answer cannot be $0$ or $1$, ...
Exodd's user avatar
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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} ...
trion's user avatar
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31 votes
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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 ...
COTO's user avatar
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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 $...
Chris Culter's user avatar
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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 $\displaystyle\prod_{i=1}^N A_{ii}$?
Ahmed Fasih's user avatar
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Why is the Quasitriangular Hopf algebra called "Quasitriangular"?

The precise definition of a Quasitriangular Hopf algebra can be found on wikipedia. What is the reason behind the word "Quasitriangular"? Is it because the R-matrix is a triangular matrix, ...
yoyostein's user avatar
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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&...
entrelac's user avatar
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18 votes
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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 ...
Nick Alger's user avatar
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18 votes
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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 ...
Fiktor's user avatar
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15 votes
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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 $a_{ij},$ ...
Bobson Dugnutt's user avatar
15 votes
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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 &...
Keaton's user avatar
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14 votes
1 answer
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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 ...
ValJarett's user avatar
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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-...
Fernando's user avatar
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Can an orthogonal matrix move monotonically toward a signed permutation matrix?

The question is motivated by this question on Mathematics SE. Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
ryanriess's user avatar
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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 ...
M a m a D's user avatar
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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 ...
Appliqué's user avatar
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Expressing diagonal matrix using elementary matrices as generators in $\operatorname{SL}(\mathcal O_K \oplus \mathfrak a)$

Let $K$ be a real quadratic number field, $\mathcal O_K$ its ring of integers and an $\mathfrak a \subset K$ a fractional ideal. I've read in van der Geer that the group $$\operatorname{SL}(\mathcal ...
principal-ideal-domain's user avatar
11 votes
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319 views

Analytical solution for a neat semidefinite program (SDP)

Let $A \in S^{n}_{+}$ be a positive semi-definite matrix with all entries being non-negative. I wonder if there is an analytical solution to the following SDP in correlation matrix $X \in S^{n}_{+}$ $$...
zxzx179's user avatar
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11 votes
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Why can't we define traces and determinants for non-square matrices?

I'm reading a book that claims that determinants and traces are only defined for square matrices, but doesn’t really explain why. From a calculation standpoint, this seems correct because I wouldn’t ...
componentvector's user avatar
11 votes
1 answer
529 views

Determinant of $n\times n$ matrix with parameter

Problem: Let $\delta \in \mathbb{R}^+$ and $n\in \mathbb{N}$. The matrix $A_n = (a_{i,j}) \in \mathbb{R}^{n\times n}$ is defined as $$ a_{i,j} = \prod_{k=0}^{i-2}\left((j-1)\delta +n-k\right) $...
Sil's user avatar
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11 votes
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The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
Mario Carneiro's user avatar
11 votes
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475 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\...
Evan Aad's user avatar
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11 votes
1 answer
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What does the degree of a matrix minimal polynomial encode?

Let $\mathsf{F}$ be any field. Let $A$ be an $n \times n$ matrix over $\mathsf{F},$ whose rank is $r \le n.$ Let $\mu \in \mathsf{F}[x]$ be the minimal polynomial of $A.$ What does $\deg(\mu)$ tell ...
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10 votes
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Matrix powers of product of diagonalizable and orthogonal matrix

Suppose I have the following matrix constructed from some orthogonal matrix $O$ and a $\pm 1$ diagonal matrix $D=diag(\pm1,\dots,\pm1)$ $$ A = O D O^{-1} D. $$ Is there a simple way to evaluate $A^n$ ...
User71942's user avatar
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148 views

Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of ...
user avatar
10 votes
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698 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 ...
Gardenia625's user avatar
10 votes
0 answers
352 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 &...
Graz's user avatar
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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 ...
Ludwig's user avatar
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10 votes
0 answers
141 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 ...
user avatar
10 votes
0 answers
195 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 &...
Yly's user avatar
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10 votes
0 answers
253 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 ...
Gevorg Hmayakyan's user avatar
10 votes
1 answer
1k 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 ...
Christian's user avatar
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10 votes
1 answer
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Is there a name for the group of real matrices whose determinant is an element of $\pm 1$?

The group of matrices whose determinant is non-zero is called the "general linear group", and the group of matrices whose determinant is $1$ is called the "special linear group". In between these two ...
goblin GONE's user avatar
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10 votes
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450 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 $\...
mzp's user avatar
  • 1,925
10 votes
1 answer
598 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 ...
user1071136's user avatar
9 votes
0 answers
164 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 ...
A. Rex's user avatar
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9 votes
0 answers
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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 ...
Quantum-Collapse's user avatar
9 votes
0 answers
2k views

Can I go from the LU factorization of a symmetric matrix to its Cholesky factorization, without starting over?

I mistakenly computed the LU factorization and then realized that the question is asking for a Cholesky factorization, i.e., finding a lower triangular matrix L such that the symmetric matrix A has ...
User001's user avatar
9 votes
0 answers
507 views

Hadamard matrices and sub-matrices (Converse of Sylvester Construction)

Let $H$ be a $d$ by $d$ real Hadamard matrix, namely: $$HH^{T}=d I$$ where $I$ is the identity matrix and $d=2^{k}$ for some natural number $k\geq 2$. The entries of $H$ are either $1$ or $-1$ and it ...
James Smithson's user avatar
9 votes
0 answers
390 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 ...
Mats Granvik's user avatar
  • 7,396
9 votes
1 answer
289 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\\ \...
pre-kidney's user avatar
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9 votes
1 answer
385 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 ...
Maxime's user avatar
  • 206
9 votes
1 answer
3k views

Is there general formula for the exponential of a tridiagonal matrix?

For an arbitrary tridiagonal matrix of the form $$ A = \begin{pmatrix} b_1 & c_1 & 0 & 0 & ... \\ a_2 & b_2 & c_2 & 0 & ... \\ 0 & a_3 & b_3 & c_3 & ......
Tobias Kienzler's user avatar
9 votes
1 answer
244 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to \text{SL}_n(\mathbb{Z}/m\mathbb{Z})...
user avatar
9 votes
1 answer
2k 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 $A$. ...
Saal Hardali's user avatar
  • 4,779
8 votes
0 answers
152 views

Encoding primes via ranks of sign matrices

Crossposted at MathOverflow Recently I came across a very simply defined family of matrices: for $n \in \mathbb{N}$, set $A_n := (a_{ij})_{1 \le i, j \le n}$, where $$\displaystyle a_{ij} := (-1)^{\...
math54321's user avatar
  • 902
8 votes
0 answers
170 views

2 questions on Page 6 of Hoffman Kunze ( Linear Algebra)

While studying Linear Algebra from Hoffman Kunze I have following two questions : I.3. Matrices and Elementary Row Operations One cannot fail to notice that in forming linear combinations of linear ...
user avatar
8 votes
0 answers
207 views

Sum over invertible 0-1 matrices

I stumbled across the following formula when working on a research problem in theoretical computer science. I checked its correctness up to $N=5$ with a computer. I am looking for a simple proof of it....
Simon Mauras's user avatar
8 votes
0 answers
257 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 ...
user1101010's user avatar
  • 3,528
8 votes
0 answers
723 views

Von Neumann's Trace Inequality for Multiple Matrices

Von Neumann's trace inequality states that $|tr(AB)| \le \sum_{i=1}^{n} \sigma_i(A) \sigma_i(B)$ where $A, B$ are general square matrices with singular values $(\sigma_i(A)), (\sigma_i(B))$, ...
Zuza's user avatar
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