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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7 views

MATLAB - Distance/Similarity between matrices (different size)

I have many matrices that have different size. Specifically, those matrices have the same number of rows but vary in the number of column. Each row is a different signal measurements, and each column ...
0
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0answers
9 views

Notation for the ith row and column of a matrix

When noting the $i^{th}$ scalar of a vector $\mathbf{x}$ one usually does it as $x_i$, since it is a scalar When doing this for matrices that are being denoted in bold, let's say $\mathbf{A}$, how ...
3
votes
0answers
31 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...
3
votes
2answers
240 views

Binary Operations for grouping

Which of the following binary operations are closed? subtraction of positive integers division of nonzero integers function composition of polynomials with real coefficients multiplication of ...
0
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2answers
21 views

Row sum of $P^{m}$ when row sum of $P$ is $1$

Let $P$ be an $n\times n$ matrix whose row sum equals $1$. Then for any positive integer $m$ , what is the row sum of $P^{m}$ ? Now I took arbitrary $2\times 2$ matrix ...
1
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0answers
6 views

Signal recovery using Majorization-Minimization with Quadratic Upper Bound

I am trying to formulate a majorization-minimization (MM) (via quadratic upper bound) approach to total variation denoising (TVD). The total variation denoisng objective function is defined as an ...
0
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1answer
8 views

Matrix & Linear Algebra - Rows Expressed as Linear Combinations of a Set of Linearly Independent Vectors

The question arises from a proof for showing that matrices and their transposes have the same rank, in the textbook Advanced Engineering Mathematics by Erwin Kreyszig. A matrix of a certain size and ...
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0answers
7 views

Connected components of pseudospectra

In this Article, page 5 Theorem 2.3 ,what is connected components of pseudospectra of matrix polynomial?
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0answers
36 views

Retrieve the value of x,z and x

I want to learn about HOW to calculation in order to retrieve the value of x, y and x. Do you have a recommended tutorial to for a beginner in relation to linear algebra in this specific case? I ...
-1
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0answers
23 views

Eigen vectors of a matrix multiplied with its transpose [on hold]

Do the eigen vectors of $A A^T$ and $AA^T$ belong to the row, column, null or left null spaces of the matrix $A$?
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2answers
20 views

Finding eigenvalues from characteristic polynomial

I am finding it extremely hard to find the eigenvalues after finding the characteristic polynomial. For example (instead of $\lambda$ I will use $x$) I have: $-x^3+x^2+16x+20=0$, how do i find the ...
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1answer
20 views

Is the spectral radius of a Hermitian matrix a non-decreasing function of the magnitude of its entries?

I strongly suspect the answer is yes. By the min-max theorem, the largest eigenvalue of a hermitian matrix $M$ is $$ \lambda_{max}=\text{max} \left( \frac{x^*Mx}{x^*x} \right) $$ This is also its ...
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0answers
17 views

What is connected components of pseudospectra of matrix polynomial? . [on hold]

What is connected components of pseudospectra of matrix polynomial? Please see this link
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2answers
35 views

matrix with all rows positive

I am thinking about a problem in a different area than linear algebra, but I came across a matrix with sum of entries of all rows positive, i.e. a matrix $A$ such that $\sum_{j} A_{ij}>0$ for all ...
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0answers
56 views

Matrix column-wise multiplication operator

I'm trying to find the proper operator for a column wise multiplication. Consider $v=[v_1, v_2, ..., v_n]^T$ and $$A=\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\a_{2,1} & a_{2,2} & ...
1
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0answers
26 views

color conversion from RGB to YIQ

I want to convert RGB color to YIQ. AS my knowledge the formula is below: To practice this math i went a to this link Color Conversion. I enter here RGB values 32,65,32. I found the result is YIQ = ...
3
votes
1answer
36 views

Does multiplication by a positive definite matrix preserve eigenvalues?

Let $A$ be a positive definite matrix and let $B$ a matrix. Then, $AB$ is similar to $A^{\frac{1}{2}}BA^{-\frac{1}{2}}$, which is in turn similar to $B$, so I get that $AB$ and $B$ are similar. ...
0
votes
1answer
29 views

SVD decomposition of matrix

Is it correct to say that a matrix $A$ and the matrix $A^HA$ have the same eigenvectors? Proof: $$ A= U \Sigma V^* \\ A^HA= U \Sigma^2 U^H $$ Am I correct?
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1answer
34 views

Boundedness of matrix norm

Let $A$ be a n by n matrix whose entries are continuous functions of $x\in \mathbb{R}^n$. Fix a matrix norm $\|\cdot \|$ and assume that $\|A(x^\star)\| < 1$. Then, the claim is that there exists ...
0
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0answers
19 views

Strassen's Laser Method Technique AND Tensors in matrix multiplication algorithms

I understand the first algorithm presented by Strassen in 1968, for fast matrix multiplication. This was the first improvement to the naive approach of multiplying matrices. Thereafter, he went on to ...
0
votes
1answer
20 views

Spectral radius and matrix norm inequality as its consequence

I am trying to undestand a proof and there is one part that's holding me back. By assumption we have that spectral radius $\rho(A) < 1$. Hence, following inequality should hold $$\|A^k\| < C ...
1
vote
1answer
19 views

Equivalence of positivity

Let us have complex matrices and their real decompositions as $H=H_1 + \imath H_2$ and $L = L_1 + i L_2$. Further, $H_1\ge 0$ and $H_2$ is skew symmetric. $L = I - P$ where $P$ is some positive ...
5
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2answers
627 views

Largest eigenvalue of a symmetric positive definite matrix with rank-one updates

I have a $n \times n$ symmetric positive definite matrix $A$ which I will repeatedly update using two consecutive rank-one updates of the form $A' = A + e_j u^T +u e_j^T$ where $\{e_i: 1 \leq i \leq ...
1
vote
0answers
13 views

condition number with component-wise norm for the sample variance any help is appreciated! :)

I'm looking through some notes and came across the following two statements in the notes where the author states it can be shown that one leads to the next. I've tried to show this using the ...
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0answers
9 views

Singular Value Decomposition of covariance matrix

Assume that I have a random vector $${\bf h} \sim CN(0,{\bf R})$$ where $\bf R$ is the covariance matrix. Can I say that the eigen vector of ${\bf R}$ are equal to the eigen vector of ${\bf h}$? ...
2
votes
1answer
81 views

matrix with fractional exponent, not getting expected output in Matlab/Octave

I have a matrix exponential function that is called a number of times in an integration routine from the heat conduction model I'm trying to implement. It works, and my results match the samples in ...
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0answers
7 views

Solving a modulo 3 matrix system, with a constraint on the domain of the solution

Someone on cs.stackexchange suggested to post the mathematical part here, I hope I'm not crossposting. All calculations below are integer calculations under modulo 3. I am trying to solve an integer ...
0
votes
2answers
59 views

Let $A$ be a $5\times 5$ matrix all of whose eigenvalues are zero. Is $A$ symmetric, anti-symmetric, or $A=-A$?

Let $A$ be a $5\times 5$ matrix all of whose eigenvalues are zero. Which of the following are always true: a. $A=-A$ b. $A^t=-A$ (anti-symmetric) c. $A^t=A$ (symmetric) d. $A^5=0$ For b: ...
6
votes
1answer
62 views

Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row?

Let $n\geq2$ be an integer and let $a_1,\ldots,a_n\in\mathbb Z$ with $\gcd(a_1,\ldots,a_n)=1$. Does the equation ...
1
vote
3answers
38 views

Let $A\in \mathbb C$ be a $2 \times 2$ matrix, let $f(x)=a_0+a_1x+\cdots a_nx^n$ be any polynomial over $\mathbb C$. Comment on $f(A)$

Let $A\in \mathbb C$ be a $2 \times 2$ matrix, let $f(x)=a_0+a_1x+a_2x^2+\cdots a_nx^n$ be any polynomial over $ \mathbb C$. Then which of the following is true? a) $f(A)$ can be written as ...
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0answers
9 views

Simplifying an expression to matrix form

I have an equation for $i\in [1:K]$ as follows: $$y_i = {\bf H}_i {\bf w}_i s_i + \sum_{i=1, k\neq i}^K {\bf H}_i {\bf w}_k s_k $$ where uppercase bold is matrix, lower case bold is vector and ...
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0answers
10 views

$det(I+A(\epsilon))$ where $A$ is an infinite matrix and not trace class!

Assume that $A$ is an infinite matrix and it's a function of the parameter $\epsilon$. I would like to find $\epsilon$ so that the $det(I+A(\epsilon))=0$. I know if $A$ was a trace class I could use ...
1
vote
1answer
22 views

proof of matrix positive semi definite

I have question about the proof about positive semidefinite (p.s.d) of a matrix. Let's say $M$ is a d by d p.s.d matrix, $H$ is any d by n matrix with n larger(or much larger) than d. Then how about ...
2
votes
2answers
28 views

LU-factorization: why can't I get a unit lower triangular matrix?

I want to find an $LU$-factorization of the following matrix: \begin{align*} A = \begin{pmatrix} 3 & -6 & 9 \\ -2 & 7 & -2 \\ 0 & 1 & 5 \end{pmatrix} \end{align*} This matrix ...
0
votes
2answers
19 views

Is this matrix multiplication correct

Having two diagonal matrices $A$ and $D$ where ${\bf a}_i$ is a $M\times 1$ vector for $i\in[1:n]$ while $d_i$ is a scalar. $$A = \begin{bmatrix} {\bf a_1} & 0 & 0 & \dots &0 \\ ...
3
votes
0answers
94 views

Transpose of the adjacency matrix

As homework I had to do an adjacency matrix for the following graph: My solution was the following: $$ \begin{bmatrix} 0&0&1&0&0 \\ 1&0&0&1&0 \\ ...
0
votes
1answer
4 views

Solving Augmented Matrix Breaking Strict Triangle Form

I'm trying to solve the following system of equations: $ 3x_1 + 2x_2 + x_3 = 0\\ -2x_1 + x_2 -x_3 = 2\\ 2x_1 - x_2 + 2x_3 = -1 $ From which I'm using the augmented matrix: $$ \left[ ...
0
votes
0answers
5 views

Concentration bounds on Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...
2
votes
3answers
74 views

If square matrix A satisfying $A^2-4A+4I=0$ does it follow that A is diagonizable?

I am given the following statement and asked to determine whether it is true or false: If A is a n x n matrix, and $A^2-4A+4I=0$, then A is diagonizable. Any help is appreciated, thank you.
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1answer
19 views

A question about matrix kernels and Kronecker products

Let us define $$ v:=v_A\otimes v_B\quad (*) $$ where $v_A$ is a fixed vector in $\mathbb{R}^{d_A}$, $v_B$ is any vector in $\mathbb{R}^{d_B}$ and $\otimes$ denotes the Kronecker product. To rule out ...
2
votes
1answer
40 views

Frobenius norm of product of matrix

The frobenius norm of matrix $F$ with dimension $m\times n$ is defined as $$||F||^2_F = \sum_{i=1}^m\sum_{j=1}^n |f_{i,j}|^2$$ If I have the multiplication of two matrices $$FG$$ where G is matrix ...
2
votes
1answer
32 views

How to compute the matrix $S$ in Sylvester's law of inertia

Sylvester's law of inertia states that for any symmetric matrix $A$ there exist an invertible matrix S such that, $S^T A S = D$, where $D$ is a diagonal matrix which has only entries 0, +1 and −1 ...
0
votes
1answer
9 views

Isometries and Orthogonal Matrices

I know how to show that multiplying by an orthogonal matrix preserves the angle and distance between two vectors. I have seen everywhere that Orthogonal matrices are kind of related to rotations and ...
2
votes
1answer
22 views

Matrix representations of tensors

I've been trying to teach myself general relativity, and I always get stuck at the same point: I don't really understand what the metric tensor is. Unless I'm incorrect, and please correct me if I'm ...
1
vote
1answer
60 views

Exponential of 4x4 matrix

It is asked to calculate $e^{tA}$, where $$A=\begin{pmatrix} 0&1 & 0&0 \\ 3\omega ^2&0 &0 &2 \omega \\ 0& 0 & 0 &1 \\ 0& -2 \omega &0 ...
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0answers
32 views

Abuse of notation ? $(A\mid M_{n\times p})$ to denote a set of matrices…

Let $A\in M_{n\times m}$. Would it be considered an abuse of notation to write $$\left(A\mid M_{n\times p}\right)\subseteq M_{n\times (m+p)},\tag{1}$$ where $\mid$ denotes matrix augmentation ? By ...
1
vote
2answers
38 views

Explicit example of a basis of invertibles for $n\times n$ matrices

Using a topological (+linear algebra) argument, one can establish the existence of a basis spanning any square matrix using invertible matrices ( $span(GL_n (\Bbb{R}))=\mathcal{M}_n (\Bbb{R}) $). But ...
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0answers
29 views

Eigenvalues of Moore–Penrose Pseudo-Inverse of a Symmetric Matrix

I was wondering if there is any bound or inequality for the eigenvalues of Moore–Penrose pseudo-inverse of a real $n\times n$ symmetric matrix $A$ in terms of eigenvalues of $A$, namely $\lambda_i$'s ...
2
votes
2answers
43 views

Computing matrix exponential of non-diagonalizable 2x2 matrix

Compute $e^M$ where $M=\begin{bmatrix}8 & -1\\4 & 4\end{bmatrix}$ Because M is not diagonalizable i try to use Jordan decomposition so i find the Jordan matrix to be $J=\begin{bmatrix}6 & ...
0
votes
3answers
33 views

matrix times its transpose equals minus identity

What would be a good example for a $n\times n$ matrix such that $A^{T}A=-I$? It would be better if you can give a matrix which has a well-known name (like "rotation matrix" etc). Thanks!