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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1
vote
2answers
56 views

About diagonalization

"Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
2
votes
1answer
453 views

Product rule when differentiate matrix products

I want to differentiate the following expression with respect to $b$ $(Y-Xb)'(Y-Xb)$ Where $Y$ is $n\times1$ and $X$ is $n\times k$ and $b$ is $k\times1$, ' denotes transpose. If i do it term by ...
-4
votes
1answer
32 views

Determine the matrices $A$ and $B$

If $$ 2A + 3B = \begin{bmatrix} 8 & 3 \\ 7 & 6 \end{bmatrix} $$ and $$ A + B^T = \begin{bmatrix} 3 & 1 \\ 3 & 3 \end{bmatrix} $$ Determine the $2\times 2$ matrices $A$ and $B$.
72
votes
0answers
2k views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
22
votes
0answers
977 views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
12
votes
0answers
174 views

determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
12
votes
0answers
182 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 ...
11
votes
0answers
303 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
430 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
168 views

Conceptual proof relating linear fractional transformations to matrices

Define a map from $2 \times 2$ invertible matrices to linear fractional transformations $$ f:\left( \begin{array}{ccc} a & b \\ c & d \\\end{array} \right) \mapsto \frac{az + b}{cz + d}.$$ ...
8
votes
0answers
146 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. ...
7
votes
0answers
184 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
139 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 ...
7
votes
0answers
65 views

Intuiting Product of Elimination Matrices (and NOT by Matrix Multiplication)

I want to intuit, and NOT compute with matrix multiplication, $M:=\color{green}{E_{P_3 \rightarrow P_4}}\color{#CA790F}{E_{P_2 \rightarrow P_3}}E_{P_1 \rightarrow P_2},$ where: $E_{P_1 \rightarrow ...
7
votes
0answers
461 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
185 views

Determining sign(det(A)) for nearly-singular matrix A

Motivation: determining whether a point $p$ is above or below a plane $\pi$, which is defined by $d$ points, in a $d$-dimensional space, is equivalent to computing the sign of a determinant of a ...
7
votes
0answers
356 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 ...
7
votes
0answers
235 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 ...
6
votes
0answers
84 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 \\ ...
6
votes
0answers
166 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
189 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 ...
5
votes
0answers
75 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
41 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
165 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
5
votes
0answers
66 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
95 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
362 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
1k 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
79 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
233 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 ...
4
votes
0answers
97 views

How does pointwise multiplication of two matrices affect their eigenvectors?

More specifically, suppose I have a known matrix $X\in\mathbb{R}^{d\times n}$ and an unkown vector $\alpha \in \mathbb{R}^n$. What can be said about the eigenvectors of $\alpha\alpha^T \odot X^T X$ ...
4
votes
0answers
49 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 ...
4
votes
0answers
51 views

Seeking diagonal $D$ such that $AD^2B$ is positive definite (A and B are symmetric and p.d.)

Let A,B be two real symmetric, non-commuting positive definite matrices. Then the product AB is not positive definite (in the sense that ${\bf x}^TAB{\bf x}$ may be anything). I'm trying to find ...
4
votes
0answers
32 views

How to simplify a determinant

If we have a matrix like: $A_{12\times 12}=\left[ \begin{matrix} {{\Gamma }_{1}} & \mathbf{0_{5\times 6}} \\ {{\Gamma }_{2}} & {{\mathbf{G}}_{2}} \\ \mathbf{0_{5\times6}} & ...
4
votes
0answers
73 views

Expectation of the absolut value of the determinant of a random matrix

Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable $$X := |\det A|$$ Can ...
4
votes
0answers
55 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 ...
4
votes
0answers
132 views

Diagonalization

I'm having difficulty understand some questions. I will highlight the terms I do not understand. Question 1: Let $A =\begin{pmatrix} 1& -2 \\ 1& 3 \end{pmatrix}$ For the matrix $A$, ...
4
votes
0answers
29 views

Determinant of triangular matrix except for one column (atomic/Gauss/Frobenius)

Is there some "smart" way to calculate determinants that look like this? $\begin{vmatrix}-1&a_{1,2}&a_{1,3}&a_{1,4}&\cdots&a_{1,m-1}&a_{1,m} ...
4
votes
0answers
98 views

A problem on 0-1 matrices.

Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is ...
4
votes
0answers
50 views

What does the metric matrix G tell us here

Let $\phi:U \rightarrow S \subseteq \mathbb{R}^3$ be a chart from $U \subseteq \mathbb{R}^2$ to a surface $S$. $G = g_{ij}$ be the metric matrix such that $ g_{ij} = \frac{\partial \phi}{\partial ...
4
votes
0answers
57 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
4
votes
0answers
63 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 ...
4
votes
0answers
108 views

Wikipedia's Cayley Table and Pictures for 3 by 3 Permutation Matrices

Are there any explanations or clarifications of the pictures at https://en.wikipedia.org/wiki/Permutation_matrix#Permutation_of_rows_and_columns? ...
1
vote
0answers
12 views

Schur product theorem

The theorem states that the Hadamard product of two positive definite matrices $ A \circ B$ is also positive definite. Can I make any statement about a the Hadamard product of a positive definite ...
1
vote
0answers
35 views

Prove that $A$ is similar to $B$

Let $A, B \in M_n(\mathbb{F})$ such that $m_A(x) = m_B(x)$ and $f_A(x)=f_B(x)=(x-\lambda_1)^{d_1}\cdots (x-\lambda_k)^{d_k}$ for different $\lambda_1, \ldots, \lambda_k$ such that $1 \le d_l \le 3$ ...
1
vote
0answers
32 views

Prove that $V_i$ are $T$-invariant for $1\le i\le k$ and $V=\bigoplus_{i=1}^{k}V_i$

Let $T$ be a linear operator over $V$ with dim$(V)=n$ and let the ordered set $B=${$v_1,v_2,...v_n$} be a basis for $V$. Furthermore, let $A=[T]_B$ be block-diagonal matrix (that is ...
0
votes
0answers
20 views

Cubic 2x2x2 matrices (tensors), is there a special name for them?

Is there a special name for cubix 2x2x2 tensors? What properties an algebra on these matrices has?
-1
votes
0answers
21 views

Differential equation to space state excercise

This is a "back of chapter" excercise which im trying to solve, my answer doesnt match the solution printed on the book, I want to write the equation in state space matrix form without using the ...
-1
votes
0answers
23 views

Is this matrix skew-symmetric?

Is this matrix skew-symmetric? $$\begin{pmatrix}0 & 2i \\ -2i &0 \end{pmatrix}$$ It seems that this matrix is skew-symmetric and Hermitian. But eigenvalues of skew-symmetric matrices must ...
-1
votes
0answers
17 views

A theorem for positive definite matrices

I would like to show that if A and B are positive definite, then [A zeros(size(A));zeros(size(B)) B] - [A;B]inv(A+B)[A B] is positive semi-definite. I would like to extend the relation to N ...