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
6 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 ...
-4
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0answers
15 views

What is connected components of pseudospectra of matrix polynomial? .

What is connected components of pseudospectra of matrix polynomial? Please see this link
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3answers
28 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 ...
2
votes
1answer
31 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. ...
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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 = ...
0
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1answer
28 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?
2
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0answers
20 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 and let $$A=\begin{pmatrix}a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ ...
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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 ...
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0answers
12 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 ...
1
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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 ...
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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 ${\bf h}$? ...
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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 ...
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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: ...
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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 ...
0
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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 ...
0
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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 \\ ...
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 ...
6
votes
1answer
57 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
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 ...
1
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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
3answers
73 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
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 ...
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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
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 ...
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!
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0answers
25 views

Find a relation between a,b and c

$ a,b,c\in \Bbb R$ $2x_1+2x_2+3x_3=a$ $3x_1-x_2+5x_3=b$ $x_1-3x_2+2x_3=c$ if a,b and c is a solution of this linear equation system find the relation between a,b and c I dont understand the ...
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 ...
1
vote
1answer
8 views

Change of Basis in Canonical Correlation Analysis

I am studying canonical correlation analysis. And I'm completely stumped for the last few days at the following manipulation. How does the following change of basis works? The equation doesn't even ...
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 & ...
2
votes
1answer
14 views

Computation of a matrix exponential for general dimensions

Originally I wanted to prove something else then I hit on this question that I find quite interesting but I don't know how to prove it elegantly. Let $$J=\begin{pmatrix} 0 & I \\ -I & 0 ...
1
vote
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 ...
0
votes
1answer
26 views

Union of subspace

Q. Say U and W are subspaces of a a finite dimensional vector space V (over the field of real numbers). Let S be the set-theoretical union of U and W. Which of the following statements is true: a) ...
0
votes
1answer
35 views

Is this “truncating” matrix function well known?

I'm working with a kind of "truncating" matrix function $\tau_r:M_{n\times n}\to M_{n\times r}$, where $r\leq n$, defined by $\tau_r(A)=B$, where $b_{ij}=a_{ij}$ for $j\leq r$. Is this a well known ...
1
vote
2answers
44 views

Eigenvectors and Kronecker product

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
0answers
27 views

What do we call the result of wedging together the columns of a matrix?

We can wedge together the columns of a square matrix to compute its determinant. More generally, the exterior product of the columns of a $b \times a$ matrix tells us the determinant of each $a \times ...
0
votes
1answer
44 views

Matrix derivative of a special function

I need some help for calculating the matrix derivative of a special function. I have checked Wikipedia and Matrix Cookbook, but could not get the answer or idea. Let us define $f(X)$ as ...
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votes
0answers
20 views

Compute the operator norm of the linear transformation defined by the following matrix. [on hold]

Compute the operator norm of the linear transformation defined by the following matrix. \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}
1
vote
2answers
38 views

How can you expand the adjoint of a matrix into a polynomial with matrix coefficients?

This book contains an algorithm which claims that a matrix $sI - A$, where $A$ is some $n \times n $ square matrix and $s$ a variable can be expanded into $$adj(sI - A) = K_0 s^{n-1} + K_1 s^{n-2} ...
2
votes
2answers
70 views

How to derive this matrix equation

$$ \left< Z,X-L-S \right> \quad +\quad \frac { r }{ 2 } \left\| X-L-S \right\|_F^2 \quad =\quad \frac { r }{ 2 } { \left\| L-\left( X-S+\frac {Z}{r} \right) \right\| }_F^2 $$ I think $ ...
-1
votes
2answers
45 views

If $A,H\in GL_{n}(\mathbb{R})$, What happen for $H A H^{-1}$, if $h(i,j)\longrightarrow‎ 0$?

If $A$ and $H$ is a two $n\times n$ matrixs, such that $\det(A)\neq 0$ and $ \det(H)\neq 0$, what happen for $H A H^{-1}$, if $h(i,j)‎\longrightarrow‎ 0 $? Is $H A H^{-1}‎\cong A$?$ \ \ \ (1\leq ...
0
votes
1answer
26 views

General Element of U(4)

Relating back to previous question about how to write a general element of $U(2)$, I am now wondering about how to write a general element of $U(4)$. Define $\Gamma_{(i,j)}:=\sigma_i\otimes\sigma_j$ ...
1
vote
1answer
28 views

Row equivalence implies independent columns?

I need to prove that "given" two matrices are row equivalent, a set of columns of the first matrix are linearly independent iff the corresponding columns of the second matrix are linearly independent. ...
1
vote
1answer
13 views

Linear transformation with matrices in base

Consider the vector space of real $2 x 2$ matrices and take as base $\{{E_{11},E_{12},E_{21},E_{22}}\}$. Where $E_{ij}$ represents the matrix with a $1$ in the $i$-th row and $j$-th column and the ...
3
votes
2answers
40 views

What is an Eigenbasis and how do I calculate it with the information below.

I have the matrix $$A = \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4 \end{bmatrix}$$ I've calculated the Eigenvalues and Eigenvectors as follows with help in a previous ...
4
votes
1answer
39 views

Example of an nonidentity element in the kernel of the map.

This question is related to my previous question here. Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: ...
1
vote
3answers
19 views

Finding the rank of an non-invertible matrix

I have a $3\times3$ matrix with three different eigenvalues $0,1, 2$. The question is: what is the rank of this matrix? If the matrix was invertible, I could say that the rank was equal to $n=3$. ...
0
votes
0answers
17 views

Why do the diagonals of a matrix have to be greater than 0 for the matrix to be positive definite? [duplicate]

Why do the diagonals of a matrix have to be greater than 0 for the matrix to be positive definite? Please provide an example (with numbers if possible).