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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0
votes
2answers
34 views

$f$ is a differentiable map and compute $Df(A)(H)$.

Let $f : GL(n, \Bbb R) \to GL(n, \Bbb R)$ be defined by $f(A) = A^{-1}$ where derivative of the matrix $A$ exists. Then $f$ is a differentiable map and compute $Df(A)(H)$. $A A^{-1} = I \implies ...
4
votes
1answer
24 views

Exist basis, simultaneously upper-triangular?

Let $A, B \in M_n(\mathbb{C})$ be such that $\text{rank}(AB - BA) \le 1$. Does there exist a basis of $\mathbb{C}^n$ with respect to which $A$ and $B$ are simultaneously upper-triangular?
1
vote
1answer
44 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
1
vote
1answer
20 views

Equivalence of positivity

Let us have complex matrices and their real decompositions as $H=H_1 + \imath H_2$ and $L = L_1 + i L_2$. Further, $H_1\ge 0$ and $H_2$ is skew symmetric. $L = I - P$ where $P$ is some positive ...
1
vote
1answer
27 views

relation of dim kers of AB and B operators

I try to prove For any matrixes $A_{ms},B_{sn}$ $$\operatorname{rank}{A}+\operatorname{rank}{B}-s\leq\operatorname{rank}{AB}$$ First, as for any $X$ that $BX=0$ also $ABX=0$, that ...
1
vote
1answer
454 views

Diagonally Dominant Matrix Preserved after Gaussian Elimination (with a modification)

Prove or disprove: If a matrix has the property $0 \neq |a_{ii}| \geq \sum_{\substack{j=1 \\ j \neq i}} |a_{ij}| $ then Gaussian Elimination (without pivoting) will preserve this property. I assume ...
0
votes
1answer
17 views

Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
0
votes
1answer
10 views

Is this relation considered antisymmetric and transitive?

I'm having trouble understanding whether or not this relation would be considered antisymmetric and transitive. The a relation R on the set of real numbers by (x,y) ϵ R if and only if x-y=0. If I am ...
0
votes
1answer
23 views

How does permutation works in “multimatrices”?

I want to adequately define a $m\times n$ "multimatrix" that satisfies these properties: 0.A $m\times n$ multimatrix has $m\times n$ entries just like a normal matrix. It is the positions they occupy ...
0
votes
1answer
17 views

How to find all square Hermitian matrices of a given dimension?

My question has a couple of parts. First off, I'm interested in finding ALL possible n x n Hermitian matrices for a given n > 2. Secondly, I'd like to find those matrices whose eigenvalues are $\pm ...
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
296 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
385 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 ...
9
votes
0answers
130 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
244 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
103 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 \\ ...
3
votes
0answers
34 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...
2
votes
0answers
20 views

Problem with determinant

Let $A\in\mathbb{C}^{3\times 3}$ and $x,y\in\mathbb{C}^3$. Prove that $det\left(I-\frac{xy^*A}{1+y^*Ax}\right)=\frac{1}{1+y^*Ax}$ How can I prove this?
2
votes
0answers
15 views

Matrix product bound

Consider the following inequality \begin{align*} -AB^{-1}A^\top \preceq cI \end{align*} where $A\in\mathbb{R}^{n\times m}$, $B\in\mathbb{R}^{m\times m}$, $c\in\mathbb{R}$ (given), and $I$ is the ...
2
votes
0answers
8 views

Matrices with left and right singular vectors being vandermonde matrices

Assume we have matrices ${\bf H_i}$ for $i\in[1:K]$ and that the Singluar Value Decomposition (SVD) of ${\bf H_i}$ is such that $${\bf H_i = A_{bi} D_iA_{si}^*}$$ where ${\bf A_{bi}}$ and $ {\bf ...
2
votes
0answers
80 views

Row-normalized and column-normalized matrix notation

I'm searching for the mathematical, algebraic notations of a row-normalized and column-normalized matrix. For example, let us consider the following matrix A: $$ A = \begin{pmatrix} 2 & 7 \\ 4 ...
1
vote
0answers
8 views

Question on applications using schur complements

i wonder if you may be able to contribute some areas/ideas where the use of schur complements are used. Like for exampple, I think schur complements can be used to check for positive definiteness of ...
1
vote
0answers
11 views

Signal recovery using Majorization-Minimization with Quadratic Upper Bound

I am trying to formulate a majorization-minimization (MM) (via quadratic upper bound) approach to total variation denoising (TVD). The total variation denoisng objective function is defined as an ...
1
vote
0answers
34 views

Need help finding elementary lower triangular matricies

Given the matrix $$A= \left( \begin{array}{ccc} 1 & 2 & 1 \\ -1 & 1 & 2 \\ 2 & 2 & 4 \end{array} \right) $$ I need to find the elementary lower triangular matricies ...
0
votes
0answers
1 view

Matrices of Ordered Bases

Let $V$ be a real finite-dimensional vector space and $T : V → V$ be a linear map. Let $E$ be a basis of V . What does it mean to say that $A$ is the matrix of $T$ with respect to $E$ and let $S : V → ...
0
votes
0answers
10 views

Sign of $tr(A)$ given $I_n+A+A^2+A^3=0$

Let $A$ be a matrix such that $I_n+A+A^2+A^3=0$, what is the sign of $tr(A)$ ($tr$ being the trace) ? What I have done : One can easily figure our the inverse of $A$ since ...
0
votes
0answers
8 views

Augmented Matrix and Row echelon form

For which real numbers s and t does the following linear system have (a) no solution, (b) exactly one solution, or (c) infinitely many solutions? Justify your answers. (s−1)x +(s+3)y + z = 1 s x ...
0
votes
0answers
20 views

Is there any smart way to check triangle inequality for a matrix?

Here is the description of the problem: We have a matrix with: all (i,i) cells are 0; Some cells are filled with certain number while others are left blank. Now, we want to fill the blanks with ...
0
votes
0answers
15 views

cofactor expansion

I want to find the determinant of the following matrix using cofactor expansion: ${ \begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 7 \\ 6 & 8 & 9 \\ \end{matrix} }$ So I am going to use ...
0
votes
0answers
7 views

Eigen-decomposition of augmented block rectangular matrix

I have a rectangular matrix $\mathbf{X}_{n\times p}$ where the eigenvector decomposition of its inner product with itself is $$ \mathbf{X}^T\mathbf{X} = \mathbf{P}^T\mathbf{\Lambda P} $$ where ...
0
votes
0answers
12 views

Geometrical interpretation of the condition number as measure of matrix dissimilarity

Consider two $p$ by $p$ symmetric positive definite matrices $\pmb F$ and $\pmb G$ and denote $$\pmb D=\pmb G^{-1/2}\pmb F \pmb G^{-1/2}.$$ Sometimes, the condition number of $\pmb D$ will be used ...
0
votes
0answers
13 views

Tensors, indices and matrix notation - is there a common convention?

For a tensor named T with two indices, there are four possibilities: $T_{ij}$ , $T_i^{\ j}$, $T^i{\ _j}$ and $T^{ij}$. Is there a common convention as to how these tensors would be represented as ...
0
votes
0answers
11 views

Notation for the ith row and column of a matrix

When noting the $i^{th}$ scalar of a vector $\mathbf{x}$ one usually does it as $x_i$, since it is a scalar When doing this for matrices that are being denoted in bold, let's say $\mathbf{A}$, how ...
0
votes
0answers
14 views

Connected components of pseudospectra

In this Article, page 5 Theorem 2.3 ,what is connected components of pseudospectra of matrix polynomial?
0
votes
0answers
13 views

Matrix & Linear Algebra - Rows Expressed as Linear Combinations of a Set of Linearly Independent Vectors

The question arises from a proof for showing that matrices and their transposes have the same rank, in the textbook Advanced Engineering Mathematics by Erwin Kreyszig. A matrix of a certain size and ...