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
55 views

Hermitian Matrix with their eigenvalues arranged in non-decreasing order

I need to formulate one property of Hermitian Matrices. It goes like this; If A, B $\in M_n$ are hermitian and their eigenvalues are arranged in non-decreasing order , then $\lambda_i(A+B)\leq ...
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1answer
19 views

Projection on cone of non-negative definite matrices

Ok, so if you have a real symmetric matrix $Q$ then the projection of that matrix on the cone of symmetric non-negative definite matrices $\mathcal{C}$ can be explicitly found if we do an ...
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1answer
31 views

How to show these two matirces are similar?

How can I show that these two block matrices are similar? $M_1 = \begin{bmatrix} AB & 0\\ B & 0\\ \end{bmatrix}$ and $M_2 = \begin{bmatrix} 0 & 0\\ B & BA\\ \end{bmatrix}$ where ...
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2answers
37 views

Find the axis of rotation from the rotation matrix.

This is a problem from the book "Mathematical Methods in the Physical Sciences" Third Edition by author Mary L. Boas. on page 129, Example 5, just in case any of you are familiar with it. So I ...
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0answers
31 views

Mapping unit sphere to ellipsoid

Consider an $N$-dimensional space. Let $M$ be a square $N\times N$ (real, but I am interested in complex case too) matrix. Are the following (hyper)ellipsoids (or degenerate hyperellipsoids)? $\{v ...
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2answers
19 views

orthonormal vector properties

I have noticed a matrix property that is outlined below: I have a set of n orthonormal eigenvectors that form a basis in Rn. If these vectors are combined to form an nxn matrix where each column is ...
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0answers
37 views

Matrixes and modulo of a vector

Consider an $N$-dimensional space. Consider the function $\kappa$ which maps a square $N\times N$ matrix $M$ into the scalar field $v\mapsto \lvert Mv \rvert$ (for $v$ being a vector). Is the ...
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2answers
167 views

Matrix multiplication question (diagonal matrices)

Suppose $AB = BA$ and $A^2+B^2 = I$, where A and B are complex matrices. My feeling is that this implies that both A and B are diagonal matrices. But I'm having trouble proving it. Appreciate any ...
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1answer
30 views

JCF of matrices $A^2$ and $B^2$

Going through a past paper and I've come across this True or False question: If $A$ and $B$ have the same Jordan Canonical Form (JCF), $A^2$ and $B^2$ have the same JCF. I thought it was true, and ...
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1answer
28 views

Expressing a matrix in terms of subgroup generators using Magma

With Magma, it is possible to define a subgroup $H$ of a finite matrix group $G$ in terms of generators. Given a matrix $M\in G$, Magma can also determine whether $M\in H$. Presumably, if Magma ...
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1answer
16 views

Spectral Norm of $2\times 2$ symmetric matrix

Consider a $2\times 2$ symmetric matrix, in this case, is there some closed formula for its spectral norm ? By spectral norm I mean the induced 2-norm, there is a definition here. Thanks.
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1answer
49 views

How to find all $3\times3$ matrices $A$ that satisfies $A^2-3A-4I = 0$? [on hold]

How to find all $3\times3$ matrices $A$ that satisfies $A^2-3A-4I = 0$?
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2answers
25 views

Dot product with vector and its transpose?

I'm having trouble with the statement: $$||\textbf{v}||^2=\textbf{v}\cdot\textbf{v}=\textbf{v}^T\textbf{v}$$ taking $\textbf{v}$ as a column vector in an orthogonal matrix. How can you do the dot ...
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0answers
23 views

Gerschgorin Theorem singularity proof

I know how to prove the Gerschgorin Theorem but how exactly would one show that there are no values of $\mu$ s.t. $\mu<0$ for which $A-\mu B$ is singular where $$ A= \begin{bmatrix} ...
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2answers
64 views

Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.

Let A,B real or complex matrixes. Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$. I demonstrated the reciprocal: $\Leftarrow )$ The two equations are ...
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1answer
36 views

Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues - Proof Strategy [Lay P397 Thm 3]

Herein, I denote the Hermitian conjugate by * (ie: $A* = \bar{A}^T) $. Let $v_i$ and $v_j$ be two eigenvectors of an Hermitian matrix H. First of all suppose that their respective eigenvalues i and j ...
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2answers
52 views

For which $a$ is a matrix $A$ diagonalizable?

Say I have a matrix $A_a$ with $$A_a:= \left(\begin{array}{c} 2 & a+1 & 0 \\ -a & -3a & -a \\ a & 3a+2 & a+2 \end{array}\right)$$ I was wondering if there was an ...
0
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1answer
21 views

Determinant of a square matrix with main diagonal of zeros?

How can I show that the determinant of a square matrix A of dimension NxN with all elements equal to $-\delta$ except the main diagonal composed by zeros, is equal to $-(N-1)\times \delta^N$?
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0answers
37 views

Matrix similarity problem (complex, real)

I'm trying to solve this problem: Given complex matrices A and B, prove there's a nonsingular real matrix Q such that $A=QBQ^{-1}$, if and only if there's a nonsingular complex matrix S such that ...
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1answer
51 views

Minimzing the generalized dissimilarity measure

I am trying to solve the following problem for quite some time now, but with no progress. Here is the problem. Let $x_1....x_n$ be n samples in d-dimmensional space and let $S$ be a non ...
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0answers
13 views

Asymmetric n*n positive matrix factorization methods?

It's better that original space does not rotate (SVD rotates the axis). I think cholesky decomposition is nearly possible but it requires the matrix should be symmetric. Some suggestions?
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2answers
30 views

Significance of an eigenvector being equal to a unit vector?

I was reading ahead in my math book when I came across a matrix denoted as A = $\begin{bmatrix} 1 & -1 & 0\\ 2 & -2 & 0\\ 6 & 0 & -2\\ \end{bmatrix}$. I then found the ...
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1answer
39 views

Find Determinant of A

I've tried creating a triangular matrix, tried row reducing but can't figure it out as I keep on having c-unknown in my answer. How would I do this?
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1answer
36 views

How to find the inverse of this particular symmetric matrix

Basically, I have a $n \times n$ symmetric matrix, which looks like this: $$ \begin{bmatrix} 1 & \alpha & \cdots & \alpha \\ \alpha & 1 & \cdots &\alpha \\ \vdots &\vdots ...
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1answer
29 views

The inverse of a matrix (main diagonal $2$, left and right of it $-1$)

I want to find inverse matrix of the ...
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2answers
103 views

Proof: $Ax=x$ for all $x$ implies $A=I$ [on hold]

Let $A$ be a square matrix of order $n$ and let $x$ be an $n$-vector. Prove that if $Ax=x$ for all $x$, then $A=I$. Thanks in advance
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4answers
83 views

Prove the matrix satisfies the equation $A^2 -4A-5I=0$ [on hold]

How to prove that $$ A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} $$ satisfies the equation $A^2 -4A-5I=0$?
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0answers
32 views

Markov Transition Matrix

I have some data, shown below. How do I construct a transition matrix, for Markov Chain ? I need the formula to calculate observation data into transition matrix. Thanks! Accumulative ...
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0answers
14 views

Is there a quickest method for computing QR decomposition by hand?

I'm currently studying QR decomposition and I've seen that one can arrive at $A=QR$ via the Gram-Schmidt process, using Householder matrices or using Givens rotation matrices. My question is: when ...
2
votes
2answers
52 views

solving $X^2 - 3X - A = 0$ where $A,X \in \mathbb{M_2(\mathbb{R})}$

Given $A = \begin{pmatrix} 7 & 3 \\ 3 & 7 \end{pmatrix}$ find a $2\times 2$ matrix $X$ s.t. $X^2 - 3X - A = 0$, in the previous parts I have diagonalised $A$ and got $P^{-1}AP = \begin ...
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1answer
33 views

Differentiation of $u^{T}Su$

I want to differentiate $u^{T}Su$ wrt $u$ where $u$ is $n$ x $1$ and $S$ is $n$ x $n$matrix . So I did the following . Since $u^{T}Su$ is a number , I wrote its expression ie $$ f = ...
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0answers
31 views

What does my teacher mean by 'choosing' from a vector?

I'm revising some lecture notes from a class I missed, I'm just struggling to figure out what she means at this point. What is choosing x1=0, x2=1... etc mean? Could someone explain?
2
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0answers
28 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
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2answers
29 views

induction on matrices with powers + addition and limit

$A= \begin{bmatrix} 1-q && p \\ q && 1-q \end{bmatrix}, 0<p<1, 0<q<1,$ Using mathematical induction show that $A^n$ = $\frac{1}{p+q}\begin{bmatrix} q && p \\ q ...
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0answers
11 views

Using basic transformations to derive matrix for the reflection in a line?

Using basic transformations (translation, scaling and rotations), show all the steps to derive the transformation matrix for the reflection of points n the line : y = 3 - x I know that a directional ...
1
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1answer
18 views

A question about matrix spectrum property

Suppose $x\in\mathbb{R}^n$, $A\in\mathbb{R}^{n\times n}$. Does anyone know the answer to the following problems. (1) $\min\limits_{x\neq0} f(x)=\frac{x^\mathrm{T}A^\mathrm{T}Ax}{x^\mathrm{T}Ax}$, ...
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1answer
67 views

Why is the permanent of interest for complexity theorists?

Studying a bit about the determinant and the permanent, I'm told that although both concepts have very similar formulas, the permanent was of not much interest historically - it was until later that ...
2
votes
0answers
37 views

linear binary code problem

Let $\mathcal C$ be a $[n,k,d]$ linear binary code such that $\mathcal C$ has a systematic generator matrix $G=[I_k\mid A]$. (i) Prove that $u\in (\mathbb F_2)^k$ is coded by $c=(u\mid uA)\in ...
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3answers
23 views

How to get An eigenvalue and eigenvectors of a matrix that contain both zero column and zero row?

Could anyone help in how to get the eigenvalue and eigenvectors of a matrix that contain both zero column and zero row like : \begin{pmatrix} -1 & 1 & 0\\ 1 & -1 & 0\\ 0 ...
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0answers
17 views

The lower bound of Cheeger Inequality as the degree goes to infinity

Consider an undirected graph $G(V,E)$ with adjacency matrix $A$ and define the graph Laplacian as \begin{equation} L = D - A \end{equation} where $D$ is a diagonal matrix such that $D(i,i) = d_i$. ...
1
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1answer
22 views

Estimate $\frac{(\frac{1}{2}\lambda_{\text{min}}(A+A^T))^2}{\lambda_{\text{max}}A^TA}<1$

If $A$ is positive definite, (maybe not symmetric), how to prove that $$\frac{(\frac{1}{2}\lambda_{\text{min}}(A+A^T))^2}{\lambda_{\text{max}}A^TA}<1,$$ I know that ...
2
votes
2answers
40 views

Algorithm to compute maximum permutation sum in matrix

Given a matrix $A_{n\times n}$ of real numbers, what fast algorithms do there exist to compute the maximum value of $a_{1,\sigma(1)}+a_{2,\sigma(2)}+\ldots+a_{n,\sigma(n)}$ over all permutations ...
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1answer
38 views

Does Fermat's Little Theorem apply to matrices?

I'm working on a problem involving applying FLT to matrices, so any information about how to do this or prove this is true would be helpful. I've been doing some research and experimenting a little, ...
2
votes
1answer
30 views

Givens rotation and retraction mapping

Here's part of the texts in the book Optimization Algorithms on Matrix Manifolds, when talking about the retraction on a matrix sub-manifold. A retraction is a mapping that maps a tangent vector in a ...
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2answers
77 views

The eigenvalue of $A^TA$

If $\lambda$ is the eigenvalue of matrix $A$,what is the eigenvalue of $A^TA$?I have no clue about it. Can anyone help with that?
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2answers
39 views

Standard matrix A of T?

Help please. What would be the standard matrix of A? I know how to do number 2 and 3 but I'm just having trouble with A. I asked this earlier but I lost my account and I'm not sure if I posted ...
0
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1answer
23 views

Find unknown matrix in equation with 3 multiplications.

A matrix $D$ is calculated as $A*B*C$. I need to find the matrix $B$ given matrices $A$, $C$ and $D$. After some trial and error it seems that the following equation is needed to reproduce matrix ...
2
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1answer
19 views

Determining the structure of the abelian group, integral matrix

I am revising for my upcoming university exams and I have a past exam question that I am finding particularly challenging... a) Consider the integral matrix $$R=\begin{bmatrix} 2 & 2 & ...
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1answer
17 views

Retrieving a Matrix from a Matrix multiplication

I have made a matrix multiplication in Matlab (K, P and S are all 2x2 matrices): K = P * transpose(H)*S Now Im given K, P, and H. I need to know S. Given that ...
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1answer
12 views

Using QR decomposition to solve a system of equations with a singular matrix

If $A\in\mathbb{R}^{n\times n}$ is singular and $x,b\in\mathbb{R}^{n}$ are such that $Ax=b$, am I right in thinking that the upper triangular matrix $R$ of $A$'s $QR$ decomposition must have at least ...