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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5 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
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0answers
5 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
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2answers
55 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
32 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
7 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
9 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 ...
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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 \\ ...
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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[ ...
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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
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1answer
46 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 ...
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0answers
15 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
18 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
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3answers
66 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
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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
36 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 ...
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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 ...
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1answer
21 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 ...
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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
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2answers
41 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
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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 ...
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0answers
27 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
25 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 ...
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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
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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 ...
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1answer
39 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
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2answers
36 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
67 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 $ ...
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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
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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
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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
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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: ...
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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 $n=3$. ...
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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).
9
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1answer
59 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to ...
1
vote
2answers
50 views

What are the Eigenvectors in the following matrix?

I have the matrix A: \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4\\ \end{bmatrix} I found $\lambda I_n - A$ to be: \begin{bmatrix} (\lambda -4) & -2 & -2\\ -2 ...
0
votes
2answers
24 views

Jordan Normal Form matrix decomposition into the sum of parts that commute.

I'm learning about Jordan Normal Form matrices, and I've read that we can decompose a Jordan Normal Form matrix into the sum of two parts $$J=N_J+D_J$$where $D_J$ is the diagonal part (i.e. the ...
0
votes
1answer
65 views

Find a matrix of the linear map in the given basis

Let $Y = \{y_1, y_2, y_3\}$ be a basis for $R^3$ where $y_1 = (1, 1, 1)$, $y_2 = (4, 1, 1)$ and $y_3 = (1, 1, 2)$. Let $W = \{w_1,w_2\}$, $w_1 = (1, 1)$ and $w_2 = (2, 4)$ be a basis in $R^2$. Need ...
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0answers
14 views

Increasing a singular value [on hold]

Can any one tell me the effect of increasing one singular value (say 10 times ) larger than others.Whether it has any importance in optimization Problems .
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1answer
12 views

If the Rref(A) of a 3x3 matrix is I(A), is this a valid eigenvector?

For the vector A: EDIT: I had originally multiplied the matrix by -1. Apologies. $$ \begin{bmatrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \\ ...
1
vote
1answer
59 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 ...
0
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0answers
22 views

What does it mean to find the principal directions and radii, given a matrix?

Do I compute the SVD, getting $$U \Sigma V^*$$ and then read off the column vectors in U that correspond to the positive singular values? These column vectors span the range of the matrix A, and I ...
1
vote
0answers
31 views

minimize smallest eigenvalue

Assume $P_A,P_B$ are probability transition matrices (each element is nonnegative and row sum is 1) and $v$ is probability row vector (each element is nonnegative and sum of elements is 1). How to ...
0
votes
0answers
26 views

Help explain “3d algebra”

The following is part of my lecture note, but I get lost after the first paragraph. I know what is "even" permutation and "odd permutation" which I learned from my abstract algebra course, and figured ...