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
3answers
90 views

Is a diagonalization of a matrix unique?

I was solving problems of diagonalization of matrices and I wanted to know if a diagonalization of a matrix is always unique? but there's nothing about it in the books nor the net. I was trying to ...
1
vote
1answer
45 views

If the determinant of a matrix goes to infinity, does it means it has no inverse?

Context I have a linear time-invariant (single-input, single-output) system in state space representation (https://en.wikipedia.org/wiki/State-space_representation#Linear_systems): $$ \mathbf{x'}(t) ...
0
votes
2answers
49 views

Matrix or vector product

This is probably a simple question A factory produce a good (1) that requires 3 labor-hours in the assembly department and 1 labor-hour in the finishing department. Assembly personnel receive 19 per ...
0
votes
0answers
57 views

Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.

I would like to have a result similar to "Traces of all positive powers of a matrix are zero implies it is nilpotent". Namely: Let $R$ be a commutative $\mathbb{C}$-algebra, $A \in ...
0
votes
0answers
25 views

Bounding the off-diagonal entries of a matrix

The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $Y = \begin{bmatrix} 0 & -i \\ i & 0 ...
2
votes
2answers
32 views

How do I build this band matrix in MATLAB?

I need to build a pentadiagonal matrix in MATLAB like this: $\begin{pmatrix} 1+2\lambda & -\lambda_1 & 0 & -\lambda_1 & 0 & \cdots & 0\\ -\lambda_1 & 1+4\lambda_1 & ...
0
votes
4answers
27 views

How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$?

How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$? I tried with $\det(A - aI) = (\cos\phi - a)^2 + \sin^2 \phi = 0$ and I got somehow to $2\cos\phi = a$, and I believe ...
0
votes
0answers
10 views

Symmetry properties charge conjugation matrices in even dimension.

While reading a paper on supersymmetry (by Peter West) i faced the following problem. Its about the symmetry property of charge conjugation matrix in different space time dimension. The charge ...
0
votes
0answers
6 views

Properties of $\nabla T_k(x)\cdot \nabla T_i(x)$ for a diffeomorphism $T$

Let $T:A \subset \mathbb{R}^n \to B \subset \mathbb{R}^n$ be a smooth diffeomorphism between $A$ and $B$. Is there anything I can say about the quantity $$\nabla T_k(x)\cdot \nabla T_i(x)$$ where ...
1
vote
3answers
45 views

Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix?

If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)? I always thought that there was ...
0
votes
1answer
21 views

Rule of thumb on number of zero entries for invertibility of a $4\times 4 $ matrix?

I have to determine whether a $4\times 4$ matrix $A$ is invertible. Suppose that there are no zero columns or zero rows. Is there any rule of thumb saying how many zero entries can be at most in $A$, ...
0
votes
2answers
11 views

what is the difference between multi-linear coefficient and multiple linear regression

what is multilinear coefficient? I heard it a couple of times and I tried to google it, all I am getting is multiple linear regression. I am confused at this point.
0
votes
2answers
42 views

Find the eigenvector and eigenvalues for the following 3 x 3 Matrix?

$$ \pmatrix{5 & 8 & 16 \\ 4 & 1 & 8 \\ -4 &-4 & -11} $$ I already got the eigenvalues that is $\lambda = 1$ and $-3$. And I managed to solve the eigenvector corresponding to ...
1
vote
1answer
34 views

Orthogonal matrix $Q$ such that $\forall x\leq 0$, $Qx\geq 0$

What are the orthogonal matrices $Q$ such that for all vectors $x\leq 0$, $Qx\geq 0$? The inequality is to be understood component-wise. In dimension 1, the only possibility is $Q=[-1]$, which is a ...
0
votes
0answers
17 views

Prove two matrices are similar?

Let $G_1=[I(k), \mathcal G_1]$, $G_2=[I(k), \mathcal G_2]$, $H_1=[\mathcal H_1,I(m)]$ and $H_2=[\mathcal H_2,I(m)]$, where $\mathcal H_1, \mathcal H_2$ are transpose of $\mathcal G_1,\mathcal G_2$, ...
0
votes
0answers
8 views

Aligning matrices, normalization. Calculating coefficients.

So as a pre-task for my upcoming exam this is one of the rehearsal assignments. I can't wrap my head around this one at all, haven't seen anything like it earlier, and I can't seem to find any ...
0
votes
1answer
18 views

Can you multiply a matrix out of another one?

This is actually from a computer graphics problem. I calculate a transformation matrix by multiplying a few other ones. ...
1
vote
1answer
59 views

Prove that this $10 \times 10$ matrix is diagonalizable. [closed]

Suppose that $A$ is an non-invertible $10\times10$ real matrix, and that $\mathrm{rank}(A-3I)=7$ , $\mathrm{rank}(A-I)=4$. How do I prove that $A$ is diagonalizable?
0
votes
1answer
46 views

Does $AA^T = A^TA$ imply that A is normal?

A is $n\times n$ matrix over complex numbers. Does $AA^T = A^TA$ imply that A is normal? If not what will be a counterexample?
-1
votes
0answers
30 views

Eigenvalues of a Matrix that has differential operators as elements

Can anyone help me find the eigenvalues of the following matrix having operators as elements: $$ \begin{matrix} \frac{d^2}{dt^2} & -\omega\frac{d}{dt} & 0 \\ ...
0
votes
2answers
36 views

The number of $2\times n $ matrices in which each of {1,2,3…2n} appears once

Show that the number of $2\times n $ matrices in which each of {1,2,3.....2n} appears once and and such that each row and column is increasing is equal to the $n^{th}$ Catalan number. What i guess to ...
0
votes
0answers
5 views

Ergodicity coefficient of block matrix

I have a stochastic matrix of the following form \begin{equation} X=\begin{bmatrix}A/3&B/3&C/3\\I_n&&\\&I_n&&\\\end{bmatrix}, \end{equation} where $A,B,C$ are all $n$ by ...
-3
votes
0answers
23 views

Linear Transformation from alpha to beta [closed]

Hello I am in my Calculus 4 class and I am studying for the final and one thing I've not ever been able to understand is how to do that matrix representation. so I'm working on the practice final ...
1
vote
1answer
37 views

Finding all orthogonal matrices commuting with a positive-definite matrix

Given $M$ a symmetric positive-definite matrix, I'd like to characterise the orthogonal matrices $Q$ commuting with $M$: $MQ=QM$. $Q$ and $M$ commute if and only if they are simultaneously ...
1
vote
3answers
47 views

How to prove that A and B are similar

Let be $$A=\begin{pmatrix} \frac{-3}{2} & 2 & \frac{-1}{2} \\ \frac{-1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & -2 & \frac{3}{2} \end{pmatrix}, B=\begin{pmatrix} 0 & 1 & ...
1
vote
0answers
25 views

Find the basis $\mathcal{V}$ of $\mathbb{R}^4$ and $\mathcal{W}$ of $\mathbb{R}^3$.

Let $T:\mathbb{R}^4\to\mathbb{R}^3$ be a linear function with the transormation matrix given as: $$A=\begin{pmatrix} -3 & 2 & 3 & -3 \\ 4 & 0 & -4 & 4 \\ 2 & 0 & -2 ...
0
votes
1answer
22 views

Gauss-Jordan elimination/matrix

Hello guys i got a problem from university and i cant seem to find the answer This is the problem : ka+b+c+d=1 a+kb+c+d=1 a+b+kc+d=1 ...
3
votes
2answers
62 views

Solution of $A^\top M A=M$ for all $M$ positive-definite

I am trying to find all matrices $A$ such that for all positive-definite matrices $M$, $A^\top M A=M$. $I$ and $-I$ are obvious solutions. I can't find out it there are other such matrices and if so, ...
3
votes
1answer
27 views

Shapes described by a homogeneous quadratic equation

Suppose we have a homogeneous quadratic equation of three variables $w_1$, $w_2$, and $w_3 \in \mathbb{R}$ as follows: $$W^TAW=0.$$ where $W=[w_1,w_2,w_3]^T$ and $A$ is a non-singular $3\times 3$ ...
0
votes
0answers
11 views

Algoritm for finding new/removed colums/rows in a matrix

Question: I am looking for an algoritm for comparing two large matrixes and finding the inserted/removed rows and columns. But it can happen that cell values change. so basically suppose you have ...
-2
votes
1answer
66 views

Eigenvalues of matrix of order $n+1$

How to find eigenvalues of following matrix? $A=\begin{bmatrix} n & -1 & -1 & \cdots & -1 \\ -1 & 1 & 0 & \cdots & 0 \\ -1 & 0 & 1 & \cdots & 0 \\ ...
0
votes
2answers
32 views

Relation between eigenvectors of matrix $X^TX$ and $XX^T$

I found a surprising property of the eigenvectors of the matrix $A = X^T X$ and $B = XX^T$ experimentally. Let $X$ be $n \times d$ with $n > d$. Then $A$ and $B$ are psd matrices. The eigenvalues ...
0
votes
1answer
20 views

Basis for the space of 4*4 hermitian matrices with specific anti-commutation properties

The space of 2*2 hermitian matrices can be spanned using the basis involving identity and the three pauli matrices. Here, the pauli matrices have specific properties like: When squared they give ...
1
vote
1answer
14 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
vote
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
votes
3answers
40 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
votes
0answers
38 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
33 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
47 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
vote
1answer
39 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
25 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
vote
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
36 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
votes
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
37 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
26 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
vote
2answers
43 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 ...