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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2
votes
2answers
28 views

Let $A$ be a real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ?

Let $A$ be a square real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ? I know that real symmetric matrices are diagonalizable . Also if all the diagonal entries be ...
0
votes
2answers
45 views

Prove that elementary matrices perform row operations

How to prove that elementary matrices actually perform their intended row operations: multiplying by a constant, adding a multiple of one row to another, and switching two rows? I've seen examples ...
0
votes
2answers
33 views

The formula $\DeclareMathOperator{tr}{tr}\mathrm{adj}(A)=\tfrac{1}{2}[(\tr A)^2-\tr(A^2)]I_3-[\tr A]A+A^2$ for the adjoint of a $3\times 3$ matrix

Let $A$ be a square matrix of order $3$. Prove that $$\operatorname{adj}(A)=\frac{1}{2}[(\operatorname{tr} A)^2-\operatorname{tr}(A^2)]I_3-[\operatorname{tr} A]A+A^2$$ where $\operatorname{tr}A$ is ...
2
votes
1answer
14 views

Looking for a particular algebraic mapping from one Boolean matrix to another

Consider the following Boolean matrix: \begin{align} X&=\begin{bmatrix} 1&1&1&1&1&1&1&1\\ 1&1&1&1&0&0&0&0\\ ...
1
vote
1answer
58 views

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$ …a different approach

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way… If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is ...
0
votes
1answer
22 views

Physical meaning of cofactor and adjugate matrix

I like the way there a physical meaning tied to the determinant as being related to the geometric volume. Since the determinant can be calculated through Laplace's formula where the cofactor matrix is ...
0
votes
1answer
63 views

Matrix determinant lemma with adjugate matrix

I would like a proof of the following result, given on wikipedia. For all square matrices $\mathbf{A}$ and column vectors $\mathbf{u}$ and $\mathbf{v}$ over some field $\mathbb{F}$, $$ ...
47
votes
0answers
3k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
15
votes
0answers
211 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
219 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 ...
12
votes
0answers
368 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 ...
12
votes
0answers
436 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} ...
10
votes
0answers
119 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 ...
10
votes
0answers
551 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
186 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 ...
9
votes
0answers
198 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
72 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 ...
8
votes
0answers
101 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 ...
8
votes
0answers
143 views
+100

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 ...
8
votes
0answers
273 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) ...
8
votes
0answers
155 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 ...
8
votes
0answers
131 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 \\ ...
8
votes
0answers
268 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
201 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
475 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
292 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
170 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
159 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
75 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
238 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
336 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 $$ ...
7
votes
0answers
488 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 ...
6
votes
0answers
65 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 ...
6
votes
0answers
106 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 ...
5
votes
0answers
45 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
824 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} ...
5
votes
0answers
43 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
205 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 ...
5
votes
0answers
117 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 $B$ ...
1
vote
0answers
21 views

What is the maximum value of coefficient $f_v$ with the constraInt that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
1
vote
0answers
30 views

how to find out generators of the following free group?

following is the subgroup of SL($2,\mathbb{Z}$) \begin{bmatrix}2\mathbb{Z}+1&4\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix} how to find out its generators? i know it is free group of ...
1
vote
0answers
39 views

Show $SO_2(\mathbb{R}) \cong\{A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$.

Show $SO_2(\mathbb{R}) \cong\{ A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$, where $SO_2(\mathbb{R})$ is the group of rotations of the circle under the operation of composition. Attempt: ...
1
vote
0answers
40 views

A question about adjoint matrices

Let $T:V \to V $ be a linear map on complex vector space $V$ which is equipped with complex inner product $ <. , .> $ we know there exists a unique linear operator $T^* : V \to V $ such that ...
0
votes
0answers
8 views

Proof: dimension of the vector space of solutions to the system Bx=0

I've run into a matrix dimension proof I'm having some trouble with: Let $A,B$ be $n\times n$ matrices, and let $P(A),P(B),P(AB)$ be the vector spaces of solutions to the systems ...
0
votes
0answers
12 views

Proving whether or not a Markov chain converges

If a Markov chain is aperiodic, irreducible, and has a stationary distribution, then by the Convergence Theorem it converges to the stationary distribution. However, if the chain does not satisfy ...
0
votes
0answers
14 views

satellites attitude determination TRIAD - how are orbital reference frame vectors constructed?

I posted this same question on space.stackexchange but never received any answer. So I am posting here hoping to get an answer as this is a quite mathematical topic. I am trying to fully understand ...
0
votes
0answers
37 views

How to find a onto homomorphism between two groups?

Consider the following subgroups of $\text{SL}(2,\mathbb{Z})$ : $A$ the subgroup of matrices with determinant $1$ : ...
0
votes
0answers
24 views

Writing a Rotation Matrix About an Angle

I am asked to find a rotation matrix $R_O$ of an angle $O$ about axis $u\in R^3$, with $u$ having length of 1. I've looked up this concept on the web but I have no idea where to get started...could ...
-1
votes
0answers
21 views

$S(0,\varepsilon ) \Rightarrow F + S(0,\varepsilon ) = \left\{ {\lambda \in C:dis(\lambda ,F) \le \varepsilon } \right\}$

Let $F \subseteq {\rm{C}}$ and $S = \left\{ {x \in C:\left\| x \right\| \le \varepsilon } \right\}$. Why does $F + S = \left\{ {\lambda \in C:dis(\lambda ,F) \le \varepsilon } \right\}$? (where ...