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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39
votes
0answers
2k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
14
votes
0answers
201 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 ...
13
votes
0answers
293 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
384 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} 1 ...
10
votes
0answers
518 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 ...
8
votes
0answers
127 views

What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices ?

Given a set of points $x_1,x_2,...,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m$ x $m$ Euclidean Distance Matrix $D$ where $D_{ij}={||x_i-x_j||}^2$. We know a little bit about these ...
8
votes
0answers
177 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
8
votes
0answers
243 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 ...
8
votes
0answers
184 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. ...
8
votes
0answers
417 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 ...
8
votes
0answers
272 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 ...
7
votes
0answers
67 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 ...
7
votes
0answers
128 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
148 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
116 views

Symmetric matrix with given determinant

The matrix \begin{equation} A := \begin{pmatrix} x & 0 & 0 & z \\ 0 & y & 0 & x \\ 0 & 0 & z & y \\ y & z & x & w \end{pmatrix} \end{equation} has ...
7
votes
0answers
102 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 ...
7
votes
0answers
57 views

Does the inverse of a polynomial matrix have polynomial growth?

Let $M : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ be a matrix-valued function whose entries $m_{ij}(x_1, \dots, x_n)$ are all multivariate polynomials with real coefficients. Suppose that ...
7
votes
0answers
222 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) ...
7
votes
0answers
101 views

A very difficult problem about the existence of following $SU(2)$ matrices?

Let $G_i$ be a sequence of $2\times2$ $SU(2)$ matrices, where $i=1,2,...,n$; and $P$ represents a permutation of $\left \{ 1,2,...,n \right \}$. The question is: Does there exist a sequence of ...
7
votes
0answers
223 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
111 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 \\ ...
7
votes
0answers
484 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\ ...
6
votes
0answers
46 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But ...
6
votes
0answers
256 views

Holder inequality for matrices

I am interested in the following version of the Holder inequality. Let $D \in M_n(\mathbb{C})$ be a positive semi-definite matrix of trace $1$ and $A, B \in M_n(\mathbb{C}).$ Does it follow that $$ ...
6
votes
0answers
2k 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 ...
5
votes
0answers
50 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 ...
5
votes
0answers
38 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
98 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
52 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
101 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
63 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
91 views

mortality problem

The mortality problem is the question if some product of a given set of matrices yields the 0-matrix. In general, the mortality problem is undecidable. To have a feeling for the difficulty of the ...
5
votes
0answers
72 views

Does the cross section of $[-1,1]^n$ on a $k$-dimensional subspace always contains a rotated image of $[-1,1]^k$?

This question is inspired by a recent bounty question, but the two questions are different and solving this one, I believe, will not lead to an answer of that bounty question. Suppose $n>k\ge1$ ...
5
votes
0answers
118 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
153 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 ...
5
votes
0answers
129 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 ...
5
votes
0answers
415 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
61 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
285 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 ...
5
votes
0answers
2k 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
86 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 ...
5
votes
0answers
259 views

Finding conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$

Assume $q$ is odd. How does one go about finding the conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$? I know that for $GL_{2}(\mathbb{F}_{q})$, one can consider the possible Jordan Normal Forms of the ...
5
votes
0answers
215 views

Smith normal form of graded modules. (Major edit)

Ok, this is a major rewriting of my previous entry which no one answered. Let us have two graded $F[t]$-modules M and N with bases $m_1, \ldots, m_m$ and $n_1, \ldots, n_n$, respectively, and $F$ is ...
4
votes
0answers
73 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} ...
4
votes
0answers
55 views

Matrix which represents the product of ideal classes of 2 matrices.

Let $f(x)$ be an irreducible monic degree $n$ polynomial with $\mathbb{Z}$-coefficients and $\Theta$ be a root of $f(x)$. There is an old theorem of Latimer and MacDuffee that there is a 1-1 ...
4
votes
0answers
175 views

Conditions for Trace Inequality Tr( ( A² - B² ) Z) >= 0

Consider the $M \times M$ complex positive semidefinite matrices ${\bf A}, {\bf B}, {\bf Z}$. We have the relation $\mu_{\text{max}}{\bf I} \succeq {\bf A} \succeq {\bf B} \succ \mu_{\text{min}}{\bf ...
4
votes
0answers
71 views

What is the explicit formula (solution) to this recursively defined binary matrix?

My question concerns the following binary matrix (call it matrix $A$). Or rather the entire family of such matrices, for some number of columns $n$ and rows $2^n$. The ellipses indicate that the ...
4
votes
0answers
47 views

How to prove the following about eigenvalues

Let $\mathbf{M} = [m_{ij}]$ be a symmetric matrix of size $m\times m$ of real elements. Let $\mathbf{A} = [a_{ij}^R + ia_{ij}^I]$ be a random Hermitian matrix whose elements have variance, $\sigma^2$, ...
4
votes
0answers
143 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 ...
1
vote
0answers
8 views

condition number with component-wise norm for the sample variance any help is appreciated! :)

I'm looking through some notes and came across the following two statements in the notes where the author states it can be shown that one leads to the next. I've tried to show this using the ...