For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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57
votes
0answers
3k 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(\...
27
votes
0answers
293 views

Which polynomials are characteristic polynomials of a symmetric matrix?

Let $f(x)$ be a polynomial of degree $n$ with coefficients in $\mathbb{Q}$. There are well-known ways to construct a $n \times n$ matrix $A$ with entries in $\mathbb{Q}$ whose characteristic ...
17
votes
0answers
228 views

Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
16
votes
0answers
274 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
votes
0answers
186 views

Integrating a matrix function involving a determinant and exponential trace

I am trying to find the normalizing constant for a probability distribution and ran into a difficult integral. When $X$ is an $p \times k$ matrix, $a>0,$ and $g>0,$ how can I compute $$\int \...
14
votes
0answers
474 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} ...
13
votes
0answers
424 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 ...
11
votes
0answers
583 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 &...
10
votes
0answers
319 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix. I have already asked a (viewed but unanswered) ...
10
votes
0answers
155 views

Maximum determinant of latin squares

I strongly conjecture that the maximum absolute determinant of a latin square can be attained by a circulant matrix. For example, $\pmatrix {5&4&2&3&1 \\ 1&5&4&2&3 \\ ...
9
votes
0answers
88 views

Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
9
votes
0answers
119 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 ...
9
votes
0answers
173 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
0answers
290 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 ...
9
votes
0answers
210 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\...
9
votes
0answers
524 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-...
9
votes
0answers
310 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 ...
8
votes
0answers
50 views

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

Let $m,n \in \bf N$.Let $\bf F_p$ denote the prime field of characteristic $p$.Consider the set $$ X_m = \{A \in GL_n( \bf F_p): A^m=1 \}$$ Compute the cardinality of $X_m$. Its clear that $\...
7
votes
0answers
162 views

A conjecture on Schatten 1-norm

I have a conjecture on Schatten 1-norm. Before presenting the conjecture, let us first specify the notions used here. A matrix $A$ is said to be a density operator if $A$ is positive semidefinite ...
7
votes
0answers
81 views

What is the criterion for a matrix containing vectors and their permutations being invertible?

Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix $...
7
votes
0answers
189 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 ...
7
votes
0answers
164 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. ...
7
votes
0answers
242 views

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
7
votes
0answers
493 views

Condition of an eigenvector problem

Please, somebody help me with this problem. [Ciarlet 2.3-5] Let ${A}$ and ${B} = {A} + \delta{A}$ be two symmetric matrices with eigenvalues $$\alpha_1\ \leq\ \alpha_2\ \leq\ \ldots\ \leq\ \...
7
votes
0answers
3k views

Chain rule for matrix - i'm confused

I googled around and searched inside the forum but I'm still confused about a problem. I have 2 matrix functions $f,g : \mathbb{R}^{n \times n} \times \mathbb{R}^{a \times b} \rightarrow \mathbb{R}^{...
6
votes
0answers
188 views

Construct a matrix of polynomials to optimize condition-like score

I'm a physicist currently working on my PhD. Within my studies, my colleagues & I encountered a (strictly mathematical) problem that baffles us (and anyone else we've talked to so far) and is also ...
6
votes
0answers
268 views

Relationship between eigenvalues of two related, Euclidean distance matrices

If $X=\{x_1,\ldots,x_N\}$ is a set of points in $\mathbb{R}^n$ then one can generate a Euclidean distance matrix $D = [d_{ij}]$ where $d_{ij}=\Vert x_i-x_j\Vert_2^2$ is the square of the Euclidean ...
6
votes
0answers
97 views

Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm T}\boldsymbol\Sigma^{-1}(\mathbf{x}-\...
5
votes
0answers
71 views
+50

Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
5
votes
0answers
69 views

A conjecture concerning the irreducibility of characteristic polynomials of Arndt matrices

Letting $n \in \mathbb{N}$, let $M_{n}$ denote the $n \times n$ binary matrix with ones along the main antidiagonal and everywhere below the main antidiagonal and ones along the antidiagonal two ...
5
votes
0answers
38 views

$\text{vec}\left(A\otimes B\right)$ is not $\text{vec}\left(A\right) \otimes \text{vec}\left(B\right)$

Let $A$ and $B$ be two square matrices of dimension $a$ and $b$. $\text{vec}\left(\cdot\right)$ is the vectorization of a matrix. Now $v_0=\text{vec}\left(A\otimes B\right)$ is not $v_1=\text{vec}\...
5
votes
0answers
49 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 ...
5
votes
0answers
128 views

Proving that $\det(A) = 0$ when the columns are linearly dependent

Proposition: Let $A$ be a $(n \times n)$-matrix. If the columns of $A$ are linearly dependent, then $\det(A) = 0$. Attempt at proof: Let $A = (A_1, A_2, \ldots, A_n)$, where each $A_i$ is a column ...
5
votes
0answers
2k 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&4&...
5
votes
0answers
46 views

Matrix equivalence independent of dimension

I'm looking for a criterion on symmetric, positive definite matrices $A$ which ensures the constant $M$ in the lower bound of the norm equivalence $M\|A\|_\infty \le \|A\|_2 \le \|A\|_\infty$ does not ...
5
votes
0answers
131 views

calculation of the determinant of a block matrix little help

I need to prove $$\operatorname{det}\begin{pmatrix}A & B \\ C & D\\ \end{pmatrix}= \operatorname{det}(DA-CB),$$ where $A,B,C,D \in M_{n\times n}(R)$ with the property that $A$ and $B$ ...
5
votes
0answers
65 views

Why is it useful to know when a linear operator on a vector space is diagonalizable?

I'm currently taking a conceptual course in linear algebra, and I'm trying to understand why it would be theoretically useful or illuminating to know when a linear operator (or its matrix ...
5
votes
0answers
76 views

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
5
votes
0answers
126 views

Eigenvectors of difference of inverse matrices

I have two matrices $A$ and $B$, symmetric and positive semi-definite (in fact, they are covariance matrices), and I am interested in computing the eigenvectors of the matrix $A^{-1}-B^{-1}$. From ...
5
votes
0answers
70 views

Diagonalization of a big scary matrix

I would need to diagonalize this tridiagonal block matrix $M$: $$M = \begin{bmatrix} A & B & & \\ B^T & A & B & \\ & B^T & A & B \\ & & \ddots & \...
5
votes
0answers
217 views

Entries of a Haar distributed unitary matrix

The eigenvector matrix of a Wishart matrix is Haar distributed and that implies that the eigenvectors are uniformly distributed on a sphere. I'm interested to know what is the distribution of ...
5
votes
0answers
246 views

How to prove the positive-definiteness of a symmetric Toeplitz matrix like this?

Define a symmetric Toeplitz matrix by $$\begin{pmatrix}c_1 & c_2 & c_3 & \cdots & c_n\\c_2 & c_1 & c_2 & \cdots & c_{n-1}\\c_3 & c_2 & c_1 & \cdots &c_{...
5
votes
0answers
157 views

Complex Numbers vs. Matrix

I have a line starting at the origin, and i extend it to a point $(a,b)$ in the plane. This thing can be called a vector and be represented as $(a,b), [a\text{ }b]^T$ (column vector) or by $a\mathbf{...
5
votes
0answers
468 views

Simultaneously (generalized) diagonalizable matrices

I heard the following theorem from our textbook: Given $A,B$ are two commuting ($AB=BA$) real normal matrices. There's some real orthogonal matrix $P$ such that $P^{-1}AP$, $P^{-1}BP$ are ...
5
votes
0answers
67 views

Definition and some elementary properties of the “vector turn map”

This is actually a follow up question. I have to apologise for the length of it. I didn't anticipate to be that long. I hope that it will be proved interesting nevertheless. I used the name "vector ...
5
votes
0answers
370 views

Invariant of matrix under elementary transformations

$\DeclareMathOperator{\rank}{rank}$ Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix $$ B = \begin{bmatrix} A & b \\ b^T & c \end{...
5
votes
0answers
50 views

Classification of $n \times n$ matrix in which each two components differ in each row and column up to automorphism

Suppose, we define a class $A$ of $n \times n$ matrix as follows: $$\text{In each Row }i, \text{for any} j,k\ (1 \leq j,k \leq n )\ a_{ij} \neq a_{ik} $$ $$\text{In each Column }l, \text{for any } ...
5
votes
0answers
3k views

Possible Jordan Canonical Forms Given Minimal Polynomial

I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester. I took the polynomial to mean ...
5
votes
0answers
1k views

Symmetrizing matrix properties

A symmetrizer $P$ is a $n\times n$ symmetric matrix such that for a $n\times n$ matrix $A$ it holds that $AP=PA^T$. There exists a symmetrizer for any square matrix, and in general it is not unique. ...
5
votes
0answers
396 views

How to generate an $n \times n$ rotation matrix?

It is well known that the $2 \times 2$ rotation matrix is given by, $$\left[ \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{array} \right]$$ and ...