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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2answers
26 views

Finding eigenvalues and eigenvectors of $2 \times 2$ matix

I having a few issues finding the eigenvectors for the following matrix: $$ \begin{bmatrix} -1 & -1\\ 0 & -2 \\ \end{bmatrix}$$ I calculated the eigenvalues to be ...
0
votes
0answers
17 views

Distance/Similarity between matrices (different size) [on hold]

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
votes
0answers
18 views

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$?

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$? If this is false in general, is it possibly true for nilpotent ...
0
votes
2answers
25 views
+50

Single transformation matrix of $A \circ B$ and $B \circ A$ with certain conditions

Let $A$ is 2x1 translation matrix and $B$ is 2x2 matrix of reflection or rotation matrix (reflection, rotation, etc.). Suppose I want to find the mapping of a $y=mx+c$ line and the mapping is done by ...
0
votes
2answers
15 views

$f$ is a differentiable map and compute $Df(A)(H)$.

Let $f : GL(n, \Bbb R) \to GL(n, \Bbb R)$ be defined by $f(A) = A^{-1}$ where derivative of the matrix $A$ exists. Then $f$ is a differentiable map and compute $Df(A)(H)$. $A A^{-1} = I \implies ...
1
vote
2answers
44 views

Distinct eigenvalues and matrices problem

Let $V$ be a real vector space and $T: V \rightarrow V$ be a linear transformation. It is given that if $v_1, . . . , v_n$ are eigenvectors for distinct eigenvalues $λ_1, . . . λ_n$ then $\{v_1, . . ...
2
votes
1answer
93 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 ...
1
vote
0answers
30 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
3
votes
2answers
2k views

Matrix determinant using Laplace method

I have the following matrix of order four for which I have calculated the determinant using Laplace's method. $$ \begin{bmatrix} 2 & 1 & 3 & 1 \\ 4 & 3 & 1 & 4 \\ -1 ...
4
votes
1answer
19 views

Finding an explicit eigenvector

Let A be an nxn matrix over a field and let adj(A) denote its classical adjoint. Suppose that all column sums of A are zero so that det(A) = 0 . If rank(A) = n-1 , then any column of adj(A) ...
0
votes
1answer
16 views

How to find all square Hermitian matrices of a given dimension?

My question has a couple of parts. First off, I'm interested in finding ALL possible n x n Hermitian matrices for a given n > 2. Secondly, I'd like to find those matrices whose eigenvalues are $\pm ...
0
votes
0answers
32 views

Matrices and determinant.. [on hold]

Use elementary row operations to evaluate |A|, and then evaluate A = $$ \left[ \begin{array}{cc|c} 1&2\\ 4&5 \end{array} \right] $$ Find |(AA^T)^2| ? can anyone tell me the ...
0
votes
0answers
10 views

Geometrical interpretation of the condition number as measure of matrix dissimilarity

Consider two $p$ by $p$ symmetric positive definite matrices $\pmb F$ and $\pmb G$ and denote $$\pmb D=\pmb G^{-1/2}\pmb F \pmb G^{-1/2}.$$ Sometimes, the condition number of $\pmb D$ will be used ...
2
votes
1answer
17 views

Equality of determinants for a specific collection of square matrices of size $n=2^m$

My investigations have led me to a question that I am convinced is true. I need to show that, for a given $m$, a certain collection of square $n=2^m$ matrices have the same determinant. In dimension ...
0
votes
0answers
12 views

Tensors, indices and matrix notation - is there a common convention?

For a tensor named T with two indices, there are four possibilities: $T_{ij}$ , $T_i^{\ j}$, $T^i{\ _j}$ and $T^{ij}$. Is there a common convention as to how these tensors would be represented as ...
1
vote
1answer
452 views

Diagonally Dominant Matrix Preserved after Gaussian Elimination (with a modification)

Prove or disprove: If a matrix has the property $0 \neq |a_{ii}| \geq \sum_{\substack{j=1 \\ j \neq i}} |a_{ij}| $ then Gaussian Elimination (without pivoting) will preserve this property. I assume ...
0
votes
0answers
11 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
241 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
votes
2answers
23 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
10 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
votes
0answers
12 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
11 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
38 views

Retrieve the value of x,z and x [on hold]

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
votes
0answers
25 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 $A$?
0
votes
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 ...
1
vote
1answer
22 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 ...
-2
votes
0answers
19 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
1
vote
2answers
39 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 ...
0
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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
27 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
37 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?
1
vote
1answer
35 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
votes
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
17 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 ...
0
votes
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 ${\bf h}$? ...
1
vote
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
60 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: ...
7
votes
1answer
63 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
39 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 ...
0
votes
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 ...
1
vote
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 \\ ...