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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36
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
199 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
285 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
372 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 ...
11
votes
0answers
510 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
122 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
176 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
240 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
178 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
270 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
123 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
146 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
99 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
56 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
217 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
100 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
110 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
483 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
409 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 ...
6
votes
0answers
62 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 ...
6
votes
0answers
37 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
110 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 ...
6
votes
0answers
247 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
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
93 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
50 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
99 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
87 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
150 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
127 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
411 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
60 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
280 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
85 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
258 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
211 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
54 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
169 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
69 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
46 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
72 views

What do the eigenvalues/vectors of a metric describe?

Given a finite metric space $(X = \{ x_i \}_{i=1}^n,d)$, one can form the matrix $A$ of pairwise distances $a_{ij} = d(x_i, x_j)$. What does the eigenspectrum of this matrix say about the metric $d$? ...
4
votes
0answers
75 views

Calculation of an expression ($\max_{U}\min_i \sum_j |U_{ij}|^2 |e_i^j|^2$)

There is an orthonormal basis $\{e_i\}(i=1,\ldots,n)$ in $\mathbb{C}^n$, each of them is represented in form of column vectors $$\begin{pmatrix} e_i^1\\ \vdots\\e_i^n\end{pmatrix}.$$ My purpose is to ...
4
votes
0answers
52 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 ...
4
votes
0answers
287 views

Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
4
votes
0answers
60 views

What are some Applications of the Permanent of a Matrix?

I have a decent understanding of the determinant of a matrix in terms of its role in Telling you if a matrix is invertible (zero vs. nonzero) Expressing the product of a matrix's eigenvalues with ...