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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259
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14k views

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(\...
52
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2k 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 ...
26
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940 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} ...
25
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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 ...
21
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382 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 ...
19
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0answers
275 views

probability for a $n\times n$ matrix to have no complex eigenvalues

Let $A$ be a $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has no complex eigenvalues? The answer cannot be 0 or 1, since ...
19
votes
1answer
511 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 regular matrices in $M_n(\mathbb{F}_2)$, that have exactly $k$ non-zero entries. Is there some sort of formula to calculate $M_n^k$? $$(k < n\;\lor\;k > n^...
19
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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 ...
18
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0answers
424 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 $...
15
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7k 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&...
13
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0answers
464 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 ...
13
votes
1answer
327 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 ...
13
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0answers
534 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 ...
13
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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 &...
13
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0answers
253 views

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, ...
12
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0answers
238 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 $a_{ij},$ ...
12
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0answers
1k 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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357 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
259 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 ...
11
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0answers
343 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
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0answers
223 views

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 $...
10
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0answers
157 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 &...
10
votes
1answer
561 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 ...
10
votes
0answers
410 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
votes
1answer
471 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
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0answers
510 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 ...
10
votes
1answer
783 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 ...
9
votes
0answers
123 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 ...
9
votes
0answers
168 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 &...
9
votes
0answers
152 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
226 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
117 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
1answer
5k 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 ...
9
votes
0answers
352 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
194 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
0answers
7k 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 $\displaystyle\prod_{i=1}^N A_{ii}$?
9
votes
1answer
316 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
1k 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 $A$. ...
8
votes
0answers
347 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 ...
8
votes
0answers
243 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
343 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
168 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
104 views

Triangularization of matrix over PID

Let $R$ be a PID and let $A \in Mat_n(R)$ with all of its eigenvalues in R. Is it true that I can always find $P \in GL_n(R)$ such that $P^{-1}AP$ is uppertriangular? If so can I have a reference ...
8
votes
1answer
387 views

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 ...
8
votes
1answer
235 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

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
1answer
228 views

Exactly $n-1$ nonzero elements if $\det(A)=0$ for every arrangement

Let $x_1,x_2,\dots,x_{n^2}\in\mathbb{R}$ with the property that any $n\times n$ matrix with exactly these elements has determinant $0$. Suppose also that there are at least $n$ distinct elements. Do ...
8
votes
1answer
117 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})...
7
votes
0answers
68 views

Exponential-like power series with Fibonnaci coefficients

Background I am considering a problem (for fun, not homework!) of the exponential of a certain matrix, \begin{equation} A = \begin{bmatrix} 0 & 0 & a & b\\ 0 & 0 & c & 0\\ a &...
7
votes
0answers
137 views

An interesting puzzle in an otherwise boring game

I recently had the displeasure of playing the Xbox 360 game Tower Bloxx Deluxe. You've played a version of it if not it itself; you build towers by dropping floors on top of the increasingly swaying ...

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