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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3
votes
1answer
143 views

Order of operations of multiple Matrix Elementary Row Operations

I have two elementary row operation matrices (elimination matrices): $E_{31} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ ...
1
vote
2answers
200 views

Applying row operations to a matrix in order to get a diagonal matrix

Is there any way to obtain a diagonal matrix $D$ from a square matrix $A$ applying only row operations $R_{ij}(k)$ with $i\neq j$, $k\in\mathbb R$, which means add to the $i$-th row the $j$-th row ...
2
votes
2answers
109 views

Where did this matrix come from?

In my lecture my professor spoke of this function $R$ that takes a vector $\vec u=\left(\begin{matrix}a\\b\end{matrix}\right)$ and rotates by $\frac{\pi}{6}$ radians counter clockwise. Then he talked ...
0
votes
0answers
42 views

What is the name of the matrices which reorder edges in an adjancency so that they are closer to the diagonal? (modularity/structure)

I have seen adjacency matrices presented where the edges formed box like formations along the diagonal (not block matrices). This helped view the modularity and community structure of the group of ...
2
votes
1answer
204 views

How to show that the determinant of $A$ is non-zero?

Let $A$ be an $n$ by $n$ real matrix such that all entries not on the diagonal are positive, and the sum of the entries in each row is negative. How to show that the determinant of $A$ is non-zero?
1
vote
4answers
175 views

What is the rank of a matrix for?

I am currently working with matrices. However I know how to calculate the rank.(By calculating the the row or colume with $0$) My question is: What is the rank of a matrix for? For what can I use it ...
3
votes
1answer
78 views

Showing that $(A_{ij})=\left(\frac1{1-x_ix_j}\right)$ is positive semidefinite

Consider the matrix $A$ whose elements are $A_{ij} = \frac{1}{1-x_i x_j} $ where we have $ -1 < x_i < 1$ and $ -1 < x_j < 1$ for $ i,j=1,2,...n$. For example, when $n=3$ the matrix ...
2
votes
2answers
90 views

Matrix from eigenvalues

$A=\begin{bmatrix} a & b\\ b & c \end{bmatrix}$ Find any $(a,b,c) \in \mathbb{C}^3$ , $a \neq 0$ , $b \neq 0$ , $c \neq 0$ , so that eigenvalues of A are $\lambda_1=\lambda_2=1$ ...
0
votes
1answer
123 views

show blockmatrix is invertible

Let $B \in \mathbb{R}^{n \times n}, C \in \mathbb{R}^{m \times n}, m \leq n$ and $\operatorname{rank} C = m$ Suppose for every $v \neq 0$ with $Cv=0$ it is $v^TBv > 0$. Show: then $A = ...
4
votes
0answers
97 views

Matrix trace minimization and zeros

I would like to minimize and find the zeros of the function $$F(S,P) = trace(S-SP^{T}(A+ PSP^{T})^{-1}PS)$$ in respect to $S$ and $P$. $S$ is symmetric square matrix. $P$ is a rectangular matrix ...
10
votes
2answers
1k views

$AB-BA=I$ having no solutions

The question is from Artin's Algebra. If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions. I have no idea on how to tackle this question. I tried block ...
4
votes
3answers
157 views

Find $C$, if $A=CBC$, where $A$,$B$,$C$ are symmetric matrices.

If $A=CBC$, where $A$,$B$,$C$ are symmetric matrices and $A$,$B$ are given find $C$. $A$,$B$,$C$ are assumed to be real valued and $B$ is positive definite matrix. Does the unique solution always ...
2
votes
1answer
52 views

How to test the convexity of mutual information using leading principal minors?

I read from textbooks that the mutual information function $I(X;Y)$ is a concave function of $p(x)$ for fixed $p(y|x)$ and a convex function of $p(y|x)$ for fixed $p(x)$. I tried to test the ...
2
votes
2answers
65 views

Method for finding the number of matrices in $M_2(\mathbb{Z}_3)$ whose determinant is $1$

I just came across a question which asks to find all the $2 \times 2$ matrices in $\mathbb{Z}_3$ whose determinant is $1$. I know that since there are only three elements in $\mathbb{Z}_3$, it is ...
4
votes
1answer
60 views

Matrix calculation

The matrix is $ M= \frac{d}{d\theta} e^{A+\theta B} \mid _{\theta = 0} $ where $A$ and $B$ are both $n\times n$ matrices. I was thinking solving it by introducing the equations: $\dot x = (A + \theta ...
1
vote
1answer
113 views

Householder reflections

Let $x=\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ I want to use a Householder reflector U to keep only first element in vector x, and make everything else zero but I'm doing something wrong... ...
0
votes
0answers
42 views

Matrix equivalence relations [duplicate]

Possible Duplicate: Equivalence Relation? Column-equivalence on two $m\times n$ matrices. If $A\sim B$, how do I show that $\sim$ is an equivalence relation for all $m\times n$ matrices. ...
4
votes
2answers
75 views

how prove for n≥2 $0≤\sum_{i=1}^n \sum_{j=1}^na_{ij}≤n $ that $A=[a_{ij}]\in M_n(R)$ be real matrix that A is symmetric and idempotent

let $A=[a_{ij}]\in M_n(R)$ be a real, symmetric and idempotent matrix (i.e.$A^2=A$). how can we prove for $n \geq 2$ that $$0≤\sum_{i=1}^n \sum_{j=1}^na_{ij}≤n $$ thanks in advance.
0
votes
1answer
78 views

Bounded solution for positive-definite matrix

Suppose $A$ is a positive-definite matrix and $b$ is a vector which satisfies $ b\leq \mbox{diag}(A)$ for all entries of $b$, i.e. $b_i= b^T e_i\leq e_i^T A e_i $. The linear equations holds: $Ax=b$ ...
4
votes
1answer
183 views

What is the last index of a third-order tensor called?

In a third-order tensor I guess the first and second index would be called row and column respectively but is there a name for the third index?
0
votes
1answer
146 views

Inequality involving the trace of a matrix, and a matrix normalization

Let $U$ be a non-zero $(q\times q)$ matrix, and assume this matrix is normalized such that $tr(U^{\intercal}U)=1$. Let $R$ be a symmetric $(q\times q)$ matrix, and $N>0$ be a positive integer. ...
2
votes
0answers
120 views

A system of linear equations in integer squares - solvable?

I consider this more a "recreational math" problem, possibly lacking a solution (because it stems from the question of "magic square of squares") or simply intractable with reasonable effort. ...
0
votes
1answer
309 views

Orthogonal matrix Q of A such that $Q^T A Q$ is a diagonal matrix

Given a matrix $A$, how could I find an orthogonal matrix $Q$ such that $Q^t A Q$ is a diagonal matrix?
0
votes
2answers
206 views

Calculating Diagonal Matrix, too many zeroes in the eigen vectors, what now?

Given the Matrix $$A = \left(\begin{matrix} 1 & 1 & 0 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{matrix}\right)$$ calculate the diagol matrix $diag(A)$ Well, for this I need the ...
1
vote
1answer
95 views

Is a symmetric non-negative integral matrix with odd diagonal entries and even non-diagonal entries full rank over $\mathbb{R}$?

Let $A$ be an $n\times{}n$ matrix which satisfies the following properties: The elements of $A$ are non-negative integers. The diagonal elements of $A$ are all odd. The non-diagonal elements of $A$ ...
0
votes
0answers
452 views

Determinant of a symmetric matrix values in each column and row don't repeat

Could you help me count the determinant of this symmetric matrix? $\begin{vmatrix} a&2b&3c&6d\\b&a&3d&3c\\c&2d&a&2b\\d&c&b&a\end{vmatrix} $
1
vote
1answer
96 views

Find a $3\times 3$ matrix $X$, such that $X^{3}$ = specific matrix [duplicate]

Possible Duplicate: Given a matrix $A$ find a matrix $C$ such that $C^3$=$A$ I have stumbled upon the following question while studying for a test in linear algebra: Find a matrix $X $ of ...
3
votes
2answers
165 views

$A$ be a $n \times n$ matrix with $\text{rank}\,(A)\lt n$: multiple choice question

Let $A$ be an $n \times n$ real non-zero matrix of rank less than $n$. Then one of the following is true? : (A) there exists an $n \times n$ real non-zero matrix $B$ such that $BA = 0$. ...
2
votes
1answer
59 views

How to generate the matrix $\mathbf{M}$ with $\operatorname{cov}(\mathbf{M})=\mathbf{I}$

How can I generate a random matrix $\mathbf{M}$ such that $\operatorname{cov}(\mathbf{M}) = \mathbf{I}$ (identity). I use matlab to generate $\mathbf{M}$.
4
votes
1answer
100 views

Saturating Horn's Inequalities

If I have a matrix product of the form: $C = AB$ where $A = UDU^*$ With A and B square, Hermitian and positive semidefinite, D diagonal, U a unitary and * representing the conjugate transpose, ...
5
votes
3answers
122 views

Problem related to a square matrix

Let $A$ be an $n\times n$ matrix with real entries such that $A^{2}+I=\mathbf{0}$. Then: (A) $n$ is an odd integer. (B) $n$ is an even integer. (C) $n$ has to be $2$ (D) $n$ could be any positive ...
0
votes
1answer
119 views

Hilbert norm and Euclidean distance

For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
1
vote
0answers
66 views

All eigenvalues of matrix $A$ are real, then there exists $B$ such that $B^2=A$ [duplicate]

Possible Duplicate: Square root of a matrix Prove that if all eigenvalues of a matrix $A \in \mathcal{M} (n,n; \mathbb{R} )$ are real, then there exists $B \in \mathcal{M} (n,n; \mathbb{R} ...
0
votes
0answers
180 views

Find eigenvalue via Discrete Fourier Transform

The question is as following Let $B=\begin{bmatrix} 4 & 1 & 0 & 0 & 1 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 1 & 4 & 1 ...
0
votes
1answer
90 views

Need for algorithm on solving a set of quadratic matrix?

Firstly, I want to thank @adam W gives a good clue to solve my homework problem. I have a set of quadratic matrix need to solve(not one equation) according to the following form: ...
1
vote
1answer
174 views

set of all symmetric non-negative definite matrices are closed or not

Can anyone tell me please that set of all symmetric non-negative definite matrices are closed or not in $\mathbb{M}_n(\mathbb{R})$ with usual topology
4
votes
4answers
363 views

If $X^n$ is a diagonal matrix with distinct eigenvalues, then is $X$ also a diagonal matrix with distinct eigenvalues?

Assume that there exists an invertible matrix $P$ such that $P^{-1}X^nP$ is a diagonal matrix with distinct eigenvalues, then can I say that $P^{-1}XP$ is also a diagonal matrix with distinct ...
1
vote
1answer
66 views

Errors while calculating the unknown of a matrix?

I am currently facing a problem for calculating the unknown in a matrix: The Determinant is $A=35$ and the matrix is $$A= \begin{bmatrix} 7 & 8 & 6 & u \\ -5 & 8 & 6 ...
1
vote
2answers
40 views

Problem related to a matrix

Taking $M$ to be of the form \begin{pmatrix} a &b &c \\ d & e & f\\ g& h & i \end{pmatrix} we get (from the $2$ given conditions) $6$ equations whereas total number of ...
3
votes
1answer
63 views

Find a transformation in specified basis

My task is to find a matrix of linear transformation $\varphi$ in basis $A,B$ $\varphi:\mathbb{R}^{2}\to\mathbb{R}^{4} \varphi((x_{1},x_{2}))=(3x_{1}+x_{2},x_{1}+5x_{2},-x_{1}+4x_{2},2x_{1}+x_{2})$ ...
2
votes
4answers
686 views

What $h$ and $k$ would make this system have infinitely many solutions?

If there are an infinite number of solutions to the system $$\left|\begin{array}{cc|c} -5 & 6 & h\\ -8 & k & -7\end{array}\right|$$ then what do $h$ and $k$ equal? I know that ...
3
votes
1answer
177 views

Topological properties of symmetric positive definite matrices

Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true? (a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$. (b) $S$ is ...
0
votes
1answer
421 views

Three linked question on non-negative definite matrices.

1.a symmetric matrix in $\mathbb{M}_n(\mathbb{R})$ is said to be non-negative definite if $x^Tax≥0$ for all (column) vectors $x\in \mathbb{R}^n$. Which of the following statements are true? (a) If a ...
1
vote
1answer
273 views

Solving a Conic Matrix given these Equations

Given a conic $\Gamma$ that has the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, $\Gamma$ can be represented by the symmetric matrix $$\mathbf{C} = \begin{bmatrix} A & B/2 & D/2\\ B/2 & ...
0
votes
2answers
157 views

Matrix polynomial - is there a trick?

Is there a trick for easily solving a matrix polynomial like $$ p(A) = \left( 7\cdot A^4 - 4\cdot A^3 + 6\cdot A - 5\cdot E \right) , A = \left(\begin{matrix}2 & -1 \\ 3 & 5\end{matrix}\right) ...
2
votes
1answer
102 views

Proof of a fact about symmetric pd matrices

Several times I bumped into the following argument in my studying If $A$ is a symmetric, positive definite $n$ by $n$ matrix then there exists a nonsingular $n$ by $n$ matrix $C$ such that $A=C'C$. ...
1
vote
1answer
125 views

Proving an implication from a matrix equation

I am supposed to prove that, if $\eta$ is lorentz metric, $M$ is a 4 by 4 matrix and $x^tM^t \eta Mx=x^t\eta x$ for any column vector $x$, then $M^t\eta M=\eta$. What I did seems awfully clumsy. I ...
0
votes
1answer
152 views

To show that there exists a right and left inverse matrix for $A$

How to show that, if $A$ element of $\mathbb{C}_{p\times q}$, then there exist a left invertible matrix $B$ and a right invertible matrix $C$ such that $A=BC$? use singular value decomposition
1
vote
1answer
228 views

A set of Quadratic equation, any good algorithm?

now I'm doing my research on filter design and I'm stuck in a mathematic problems. I want to solve the following equation: ...
2
votes
1answer
80 views

Notation of Matrix and Coordination

I was confused by the notation of the following question Let $E = I_5(R_2\leftarrow R_2+4R_3)$, then $E^{-1}=I_5(C_p\leftarrow C_p+\alpha C_q)$, what are the values of $p,q,\alpha$? I know that ...