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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0
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0answers
9 views

Bounding 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 ...
0
votes
2answers
37 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
14 views

How to build a matrix in MATLAB with specific entries?

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
18 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
6 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 ...
-3
votes
0answers
20 views

Linear Transformation from alpha to beta [on hold]

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
2answers
21 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
0answers
4 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 ...
2
votes
0answers
63 views
+50

Multiplication of unitary matrices to make symmetric off-diagonal elements zero

Context Starting with a unitary matrix $U$ of size $m \times m$, I have read of a way to obtain a diagonal matrix by sequentially multiplying $U$ from the right by unitary matrices $V$ of a certain ...
0
votes
2answers
8 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
1answer
16 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 $A$, ...
1
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0answers
37 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 ...
4
votes
1answer
67 views
+50

Eigenvalue of block matrix of order $2n$

How to find eigenvalues of following block matrix? $$P=\begin{bmatrix} A & B \\ B & A \end{bmatrix}$$ Where, $A=\begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & \cdots & ...
1
vote
1answer
25 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
1answer
37 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?
0
votes
0answers
11 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
6 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
15 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. ...
0
votes
1answer
398 views

How do you calculate the dimensions of the null space and column space of the following matrix?

I understand you are supposed to get the reduced row echelon form, which I did, and this is what I came up with: 1 -2 0 19 -6 0 -37 0 0 1 -6 2 0 6 0 0 0 0 0 1 3 0 ...
18
votes
5answers
1k views

A matrix is diagonalizable, so what?

I mean, you can say it's similar to a diagonal matrix, it has n independent eigenvectors, etc., but what's the big deal of having diagonalizability? Can I solidly perceive the differences between two ...
1
vote
1answer
48 views

Prove that this $10 \times 10$ matrix is diagonalizable.

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
38 views

Changing the basis of a transformation

Given $T: \mathbb{R}^2 \to \mathbb{R}^2 : T\begin{bmatrix} x \\ y \end{bmatrix} \to \begin{bmatrix} 2x+y \\ x-3y \end{bmatrix}$ with standard basis $\mathcal{B}$ and basis ...
-1
votes
0answers
19 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 \\ ...
1
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0answers
23 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 ...
2
votes
0answers
60 views

Are two linear system equivalent? [on hold]

Let $A$ and $M$ be square matrices of size $s$ and $n$ respectively, let $k_i \in\mathbb{R^n}$ be column vectors for all $i=1,\ldots,s$. Denote $K=\left[ \begin{matrix} {{k}_{1}} \\ \vdots ...
0
votes
2answers
24 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 ...
1
vote
1answer
25 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 ...
3
votes
1answer
25 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 $3\times 3$ ...
2
votes
1answer
33 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
0answers
4 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 ...
1
vote
2answers
37 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
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 & ...
11
votes
4answers
723 views

Is there an operation on matrices such that the determinant yields a homomorphism with the additive group of the reals?

It well known that, under standard matrix multiplication $\det(AB) = \det(A)\det(B)$, or in other words, that $\det : \mathbb{R}^{n \times n} \rightarrow \langle\mathbb{R}, * \rangle$ is a monoid ...
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 ...
1
vote
2answers
39 views

Solution of $A^\top M A=M$ with $M$ 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, ...
0
votes
1answer
45 views

$\kappa(B^{-1}A)=\kappa (AB^{-1}) = \frac{\lambda_{\max}(B^{-1}A)}{\lambda_{\min}(B^{-1}A)}$

Prove or disprove: if $A,B$ are symmetric positive definite (s.p.d.) matrices then operator 2-norm condition number $\kappa(B^{-1}A)=\kappa (AB^{-1}) = ...
2
votes
1answer
25 views

Element with maximum magnitude in $A \leq \max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A

Let $A$ be a matrix with real values. Is it true that element with maximum magnitude in $A$ is less than $\max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A? That is, is $$ \max_{ij} ...
3
votes
2answers
131 views

Determinant of specially structured block matrix

How do you compute the determinant of the following $nd \times nd$ block matrix? $$M = \begin{bmatrix}A+B & A & A & \dots & A & A\\ A & A+B & A & \dots & A & ...
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
votes
1answer
71 views

Trace and transformations of a matrix

I have the following expression $$T = \operatorname{Trace}(AMA')$$ where $M$ is a square $n\times n$ matrix, and $A$ is a $m \times n$ matrix, both full-rank. The goal I want to achieve is that I do ...
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 ...
0
votes
1answer
37 views

Create matrix from image

I am strugling with a simple task: Create a matrix $A$ when you know that the image of $A$ has the basis $\langle \{ 1, 4 ,1 \}; \{ 3, 6, 2\} \rangle$ and $A(T)$ (transpose) has the image with basis ...
-1
votes
1answer
56 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
29 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
19 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 ...
19
votes
3answers
625 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 ...
1
vote
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 ...
4
votes
1answer
58 views

Is the matrix norm of a matrix equal to the maximum of the norms of its Jordan block?

Let $J$ be a Jordan block matrix with blocks $J_1,\cdots,J_n$. I came up with some examples of $J$ and noticed that $\|J\|=\max_{i=1,\cdots,n}\|J_i\|$. Does this result always hold? The norm I use ...
7
votes
3answers
6k views

Span of an empty set is the zero vector

I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone". Is Nering ...
3
votes
1answer
49 views

What is the relationship between the rank of $C_i$ and the rank of $A,B$?

Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ The ...