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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1answer
12 views

how to find the pivot/axis and angle that move one coordinates space to another?

I am writing a plugin for a 3d modeler, and I am stuck. For my plugin, I need to get the axis and the angle used for rotating a 3d object. But I only get the coordinates (~ 3dmatrices) of the objects ...
1
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0answers
42 views

Adjacency matrix of $\bar G $

Let $M$ be the all $n \times n$ matrix and $I_n$ be the $n \times n$ identity matrix. Suppose $A$ is the adjacency matrix of a simple graph $G$ on $n$ vertices. Find the adjacency matrix of $\bar ...
0
votes
0answers
50 views

Finding three 3x3 Hermitian matrices which anticommute and squares to identity.

How to find three 3x3 matrices which anti-commute and squares to identity? The best method I thought of was to take a general hermitian matrix. Find the constraints(1) on its elements such that it ...
0
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3answers
36 views

Is every invertible matrix $A$ an Adjugate matrix of some other matrix $B$? If so, is $B$ unique?

Is it true in general, true for a specific field ($\mathbb R$/$\mathbb C$) or false? could it be that $A=adj(B),A=adj(C)$ but $B\not=C$?
2
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0answers
36 views

$\det{\begin{bmatrix}\det A & \det B \\ \det C & \det D\end{bmatrix}}=0$ [duplicate]

Let $A,B,C,D \in M_n(\mathbb{R})$ and let $rank{\begin{bmatrix}A & B \\C & D\end{bmatrix}}=n$. Prove that $\det{\begin{bmatrix}\det A & \det B \\ \det C & \det ...
0
votes
1answer
32 views

Sum of the entries in the matrix $A^3$

Let $A\neq I$ be a $5\times5$ matrix with real entries such that the sum of the entries in each row of $A$ is $1$. Then the sum of all the entries in $A^3$ is 1)$\space 3$ $\qquad $2)$\space 15$ ...
0
votes
1answer
44 views

How do I rearrange this matrix equation to find A and b?

The Question: It is possible to rearrange the matrix equation $\pi^TP= \pi^T$ into a linear system $Ax = b$ where $x = \pi$ is the unique solution to the system. Such a system could be solved by, ...
1
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1answer
38 views

$t\mapsto\sin(tA)$ is continuous

How to show that $t\mapsto\sin(tA)$ is continuous for a real matrix $A\in Mat(n,n)$ Can I use trigonometric identity, $\sin y-\sin x=2\cos\left(\frac{x+y}{2}\right)\sin(y-x)$ but this holds ...
0
votes
0answers
22 views

Stability of a semidefinite programming problem

For the minimum trace factor analysis problem, I want to prove that if I change a parameter in the optimization problem, the solution will be stable. Let $\mathbf{D}^p$ denote the set of $p \times ...
1
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3answers
14 views

Find the matrix A with respect to the standard bases

Let $V=\mathbb{R}^4$, let $W=\mathbb{R}^3$ and let $\phi$ be the linear map $$\left(x,y,z,t\right)\rightarrow \left(x-2z+t,2y+z,x+4y+t\right)$$ Write down the matrix of A of $\phi$ with ...
3
votes
1answer
34 views

non abelian von Neumann algebras

I'm not familiar with von Neumann algebras, but I need the following fact (if it's true) for an other proof. Let $H$ be a Hilbert space, $A\subseteq L(H)$ a non abelian von Neumann algebra. Must $A$ ...
0
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1answer
13 views

augmented matrix of the system of linear equation

can you help me check whether my attempt to the following question. Write down the augmented matrix of the system of linear equation ...
2
votes
1answer
35 views

Exponential of a symmetric matrix

Let $A$ be a real, symmetric and positive definite matrix and suppose $B$ is a real symmetric matrix such that $\exp(B) = A$. Is $B$ unique? The solution of my homework sheet says that $B$ is ...
0
votes
1answer
26 views

How to prove the following fact regarding matrices:

I am unable to prove the following fact regarding matrices: If $A$ is a symmetric matrix then there exists a lower triangular matrix $T$ with non-negative diagonal entries such that $A=TT^t$ where ...
0
votes
0answers
25 views

QR decomposition for nondegenerate quadratic form

Let $A$ be an invertible real $n\times n$-matrix, and $q$ be a nondegenerate quadratic form on $\mathbb{R}^n$. Do we have the QR decomposition for $q$ ? In other words : is it true that there exists ...
2
votes
1answer
37 views

Find a power of matrix by Cayley-Hamilton theorem

Let $$A= \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 0 \\ \end{pmatrix} $$ And I should calculate $A^2$ and $A^{12}$ by Cayley ...
1
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2answers
38 views

Lower bound on quadratic form

Suppose I have a non-symmetric matrix $A$ and I can prove that $x^T A x = x^T \left(\frac{A+A^T}{2}\right) x>0$ for any $x \ne 0$? Can I then say that $x^T A x \ge \lambda_{\text{min}}(A) \|x\|^2 ...
1
vote
1answer
32 views

Convexity of the field of values

I am looking for alternate proofs for the convexity of the field of values. In Topics in matrix analysis by Horn and Johnson they define the field of values of an $n\times n$ matrix $A$ as $$F(A) = ...
0
votes
0answers
12 views

General form of isometric projection from $\mathbb{R}^M$ to $\mathbb{R}^N$?

Assuming $M \ge N$, I guess the projection is generally represented by the $N \times M$ matrix, and isometry would require that its "determinant" = 1? However, the determinant is not defined for ...
2
votes
1answer
19 views

Why is $\sum\limits_{k\ge0}\frac{(-1)^k}{(2k+1)!}t^{2k+1}A^{2k+1}$ absolut convergent?

Why is $\sin(tA)=\sum\limits_{k\ge0}\frac{(-1)^k}{(2k+1)!}t^{2k+1}A^{2k+1}$ absolute convergent ? for a $n\times n$ real matrix $A$ and $t\in \mathbf R$ Which crieterion is to use ?
2
votes
2answers
32 views

How do I find value of a and b in this matrix question?

This is a question from a homework sheet my teacher gave. I already did alternate a. Alternate b is quite confusing! It asks to find the value for a and b. I don't really know what to do but here's ...
0
votes
1answer
31 views

Eigenvalue of a symmetric and antisymmetric matrices

Let A be a real n×n matrix; 1) If A is symmetric, show that all eigenvalues of A (in complex numbers) are real. 2) If A is antisymmetric, show that all eigenvalues of A are pure imaginary. (They ...
3
votes
1answer
31 views

Possible ranks of a $n!\times n$ matrix with permuted rows

Let $a_1,\cdots,a_n$ be $n$ arbitrary real numbers. Form the $n! \times n$ matrix $M$ whose rows are obtained by permuting the $n$ numbers given. Find all the possible ranks of such a matrix. ...
1
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1answer
36 views

Help with understanding similar matrices

Going over past notes I do not understand the concept of similar matrices and fail to see how my lecturer has got these Eigenvectors associated to the values, I think I'm missing something can someone ...
21
votes
3answers
672 views

Sudoku with special properties

Sudoku is a puzzle, with the objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid (also "sudoku-blocks") contains all of ...
0
votes
0answers
43 views

The set of all $n\times n$ matrices A such that the $A^T = A^{-1}$ is a subspace of the vertor space $M_n(\mathbb{R})$

I think the set of $n \times n$ matrices such that $A^T = A^{-1}$ is not a vector space since it doesn't have $0$. How do I show that it's not a subspace?
0
votes
0answers
20 views

Expectation of the trace of An Inverse Wishart random matrix

Assume that Σ~IW(Α,Τ,Ν) with T>N. Σ,Α are positive definite symmetric matrices and IW stands for Inverse Wishart. What is the following Expectation? $E(tr((Σ^{-1}B)$=? let B=bb', where b is a ...
6
votes
1answer
65 views

Matrices and prime numbers

Let $ p $ be a prime number and \begin{align} K=\left\{ \begin{pmatrix} a &b \\ c& d \end{pmatrix} \mid a,b,c,d \in \left\{0,1,\ldots,p-1 \right\}, \right. & a+d \equiv 1 \!\!\!\! ...
-1
votes
0answers
17 views

Does a row echelon form have to follow the same order of zero placement? [on hold]

Can an echelon form involve zero's in different positions even if it fulfils the same requirements? Example matrix which solves like an echelon form but in different order Is this also considered ...
0
votes
1answer
10 views

The span of two subsets

The question i trying to answer is to reduce a subset to just the reduce the subset to only the linear independent elements. Also to show that the Span(X)=Span(Y). With X being the subset given and Y ...
4
votes
1answer
71 views

Calculating the determinant of a matrix using its rank

Let A, B, C and D be real n×n matrices. If $$\operatorname{rank} \begin{bmatrix} \ A & B \\[0.3em] \ C & D \\[0.3em] \end{bmatrix} = n$$ then show that $$\det ...
2
votes
2answers
66 views

If $A$ is a matrix of size $n\times n$, and $A^2+A+2I=0$

If $A$ is a matrix of size $n\times n$, and $A^2+A+2I=0$, check whether $A$ is singular or not and find its inverse if it exists. I can find the inverse by simply multiplying the given equation with ...
0
votes
0answers
25 views

Simple excercise on linear transformations - confused

A Linear tranformation L in $\mathbb R^3$ with matrix $$ L_b^b = \left(\begin{matrix} 1 & 0 & 5 \\ 0 & -2 & 2 \\ 1 & -2 & 7 \end{matrix}\right)$$ and basis $b = \{ (1,0,2), ...
0
votes
0answers
15 views

Jacobian matrix of summation function

So let's say I have a function like this $(\mu_{ij})_{i,j=1,...,t;i+j>t}\longmapsto \sum_{i,j;i+j>t} \mu_{ij}$ and I need to find the Jacobian matrix of that function. I tried to calculate it ...
2
votes
1answer
39 views

What does a homomorphism $\phi: M_k \to M_n$ look like?

Let $\phi : M_k(\Bbb{C}) \to M_n(\Bbb{C})$ be a homomorphism of $C^*$-algebras. We know that $\phi$ decomposes as a direct sum of irreducible representations, each of them equivalent to the identity ...
1
vote
1answer
40 views

Diagonalization of a Matrix Function

Is it possible to diagonalize a fundamental matrix $\phi(t)$ of a differential equation $dx(t)=A(t)x(t)dt$, i.e., is it possible to use a diagonal matrix in place of the fundamental matrix?
1
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0answers
42 views

Grid traversal algorithm

I am trying to solve a programming puzzle that goes as follows: Imagine we have a market or mall with several stores in it. The market is represented as an NxN grid where 1 <= N <= 20 and every ...
-1
votes
0answers
28 views

Eigenvalues matrices

Let matrix $$A=\left(\begin{array}{ccc} 1 & 1 & 1\\1 & \omega^2 & \omega \\ 1 & \omega & \omega^2 \end{array}\right)$$ where $\omega$ is the cube root of unity. If $a,b,c$ are ...
3
votes
1answer
19 views

“Reverse” of frobenius matrix norm inequality

Suppose that we have some $m \times n$ matrix $C$, and its full rank (skeleton) decomposition $$ C = AB^T, $$ where $A$ is $m\times r$ and $B$ is $n\times r$ for some $r$. We know that frobenius norm ...
2
votes
1answer
21 views

Orthogonal matrix representation

If $\mathbf{M}$ is anti-symmetric, then $\mathbf{U}=(\mathbf{I}-\mathbf{M})(\mathbf{I}+\mathbf{M})^{-1}$ is orthogonal with $\det\mathbf{U}=1$. This is just manipulation and noticing that ...
0
votes
1answer
27 views

Condition number for each variable

Condition number of a matrix tells us how viable it is to solve $Ax=b$ $$A= \begin{bmatrix} 1.001&1\\ 1&1 \end{bmatrix} $$ Is a matrix that would be difficult to solve numerically. However ...
0
votes
3answers
64 views

How do I calculate generalized eigenvectors?

I have the matrix $$A=\begin{pmatrix} 5 & 1 & 0\\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix}$$ and I should determine generalised eigenvectors, if they exist. I found one ...
1
vote
2answers
38 views

Problem with finding the kernel of a linear map

Let $T$ be a linear map which is represented by the following matrix in the standard basis. $$\begin{pmatrix}-1 & 0 & 1 \\ 1 & 2 & 3 \\ 2 & 3 & 4 \end{pmatrix}$$ I'm trying ...
5
votes
2answers
58 views

Show that $\left\| \exp(A)-\mathbf{1} \right\| \leq e^{\left\|A\right\|}-1$

Have been attempting this question, just wondering if my answer looks alright. Question: Given $A \in \Bbb{K}^{n\times n}$ show that $\left\| \exp(A)-\mathbf{1} \right\| \leq e^{\left\|A\right\|}-1$ ...
1
vote
1answer
48 views

Adjoint of an adjoint of a matrix

Can you please help me on this question? $\DeclareMathOperator{\adj}{adj}$ $A$ is a real $n \times n$ matrix; show that: $\adj(\adj(A)) = (\det A)^{n-2}A$ I don't know which of the expressions ...
1
vote
1answer
26 views

What is the name of the 3D matrices?

The name of a variable in the $\mathbb{R}$ is called scalar. Multiple scalars form a vector: $\mathbb{R}^n$ Two or more vectors form together a matrix: $\mathbb{R}^{n \times m }$ But what is the ...
1
vote
1answer
31 views

Numerical Range of A and A transpose.

I was playing around with the numerical range [NR] (or field of value) of a matrix $A \in \mathbb{C}^{n\times n}$ lately. And was actually looking for a proof to show: \begin{equation} A=A^H : F(A) = ...
1
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0answers
12 views

In which cases are the main diagonal elements of a product of positive definite matrices positive?

Let $A$ and $B$ be symmetric positive definite (pd) $n \times n$ matrices and $C = A \cdot B$. In which cases is then every $c_{ii}$, the $i$-th main diagonal elements of $C$, positive? When $A$ ...
0
votes
0answers
30 views

Does is matter whether you write a matrix using vector columns or rows?

given say a set of vectors in R^3. (1,1,2) , (-1,-1,3) , (6,2,2) Does is make any difference if the vectors are written as rows to form a matrix or columns? I just ask because one lecturer will write ...
2
votes
0answers
32 views

Evaluate the product $\DeclareMathOperator{tr}{tr}\tr(AB)\tr(CB^{-1})$

Let $A,C$ be given positive semidefinite matrices, $B$ be an arbitrary positive definite matrix. How can I estimate the value of $\tr(AB)\tr(CB^{-1})$ ? Is that true $\tr(AB)\tr(CB^{-1}) \geq \tr(AC)$ ...