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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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10 views

For $V = \sum_{s=1}^{t} A_s A_{s}^T$ to be non-singular $(A_s)_{s=1}^{t}$ needs to span $R^d$

I am reading a book on bandits algorithm and inside a proof it says the following: Let $(A_s)_{s=1}^{t}$ be sequence of vectors in $R^d$. Construct a matrix $V$ such that: $$ V = \sum_{s=1}^{t} A_s ...
3
votes
1answer
28 views

Are two isomorphic finite subgroups of $SO(4)$ conjugate?

Let $A,B$ be two finite subgroups of $SO(4)$ such that $A$ and $B$ are isomorphic as abstract groups. Can we find a $g \in SO(4)$ such that $$ gAg^{-1}=B? $$ If it is the case, does the same ...
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2answers
13 views

The conjugate of a scalar is the same scalar in a matrix-scalar multiplication?

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section 2.4, Page 97, I thought the equation is right as followed: $(cA)^{*} = cA^{*}$ But answer ...
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0answers
20 views

Structural controllability of networks

I want to check the structural controllability of a given network from a given input node. In Matlab, controllability can be verified using Kalman rank condition (...
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0answers
11 views

Permutative Constraint on Image Approximation

Motivation I am trying to explore the idea of constraining the approximation of an image represented by an $m$-by-$n$ matrix $A$ by the values on a linearly-spaced interval of $mn$ elements $L$ ...
4
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0answers
73 views

Matrix inverse algorithm that works for any unitary ring

Is there any algorithm to find out if a given square matrix has an inverse (which is both left and right inverse), and compute the inverse, if there is one for any unitary ring, without assuming ...
1
vote
1answer
22 views

Rewrite second order non-homogeneous differential equation as a first order system

Question: I believe I am correct up until $y(t) = Cx(t)$. I was told I did it incorrectly, but I cannot figure out how to grab the position of the antenna using $y(t) = Cx(t)$. Does it look correct, ...
2
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1answer
35 views

After multiplying a positive definite matrix several times to 'a vector A', still less than 90 degree between the 'vector A' and the 'mapped vector'?

My question Would the $\theta$ be still less than 90 degrees in vT * Mk v = ||v|| * ||Mk v|| * cos $\theta$, if the matrix M is positive definite? Background Information Let's suppose that v (...
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0answers
23 views

Extension of Choi's theorem

In the paper written by Man Duen Choi "Completely Positive Linear Maps on Complex Matrices", there was a criteria mentioned/theorem. For reference I have written it below. Let $ϕ:M_n→M_m$. Then ϕ is ...
0
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1answer
45 views

Solve $A = C B C^t$ for $B$

I know this question, but I would like to know the middle square matrix $B$. Given positive definite matrix $A \in \mathbb R^{2 \times 2}$ and non-zero matrix $C \in \mathbb R^{2 \times 3}$, find $...
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1answer
19 views

Prove that $S = \{A \in GL_n(K) | AJA^t = J\}$ is a subgroup of $GL_n(K)$

Given a field $K$, $n \in \mathbb{N}$ and $J \in K^{n \times n}$, show that: $$ S = \{A \in GL_n(K) | AJA^t = J\} $$ is a subgroup of $GL_n(K)$. To show that S is a subgroup, we must show: It must ...
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0answers
27 views

Find a basis for the kernel and image of matrix A

I need to find a basis for the kernel and image of the matrix A $A = \left( \begin{array}{ccc} -12 & 6 \\ 4 & -2\\ -8 & 4 \end{array} \right)$ But I am unsure how to do that. For the ...
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0answers
18 views

How can I find the initial state vector from state space model?

Assume that we have input $u(k)\in \Re $ and output $y(k) \in \Re$ and we estimate the black box model by using subspace identification method. $$x (k+1) = Ax (k) + Bu(k) $$ $$y (k) = Cx (k) $$ If ...
0
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1answer
33 views

Is it possible to determine the constituents of a matrix product, given the result?

Suppose we have a set $M$ of two or more matrices such that every matrix product $X$ composed of matrices drawn with replacement from $M$ is unique. Is there a set $M$ for which we can determine the ...
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0answers
32 views

Why is $ \max_{i} | \lambda_i(A) | \leq \| A \|_P $?

I was told: $$ \max_{i} | \lambda(A) | \leq \| A \|_P $$ I tried thinking through it. So the operator norm is defined as: $$ \| A \|_P = \sup_{y \neq 0} \frac{ \| A y \|_P }{ \| y\|_P } = \sup_{ \| ...
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0answers
9 views

Derivative (Jacobian) of transposed function

Let $x \in R^n$, $F \in R^{m \times n}$ and $f(x) = Fx$. It's easy to conclude that the Jacobian of $f(x)$ is $Df(x) = F$. Where $Df(x)_{ij} = \frac{\partial f_i}{\partial x_j}$. Therefore $\nabla ...
3
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0answers
56 views

Is there a name for a matrix with symmetric but inverse entries?

A matrix $A$ is symmetric if it is equal to its transpose. Then, the following equality holds between the entries of this matrix: $$a_{ij}=a_{ji}$$ Is there an established name for a similar matrix in ...
0
votes
1answer
51 views

A statement about reduced row echelon form

According to Nicholson's linear algebra : The matrix $R$ has $r$ leading ones (since rank $A = r$) so, as $R$ is reduced, the $n \times m$ matrix $R^T$ contains each row of $I_r$ in the first $r$ ...
0
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1answer
22 views

Isomorphism between $GL_2(\mathbb{F}_2)$ and $S_3$ [duplicate]

The question is to show that $GL_2(\mathbb{F}_2)$ and $S_3$ are isomorphic where $S_3$ is the symmetric group of $\{1, 2, 3\}$ and the group operation is composition. I have listed all the elements ...
0
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0answers
29 views

Partial derivative of matrix

$\renewcommand{\v}[1]{\mathrm{vec}\left(#1\right)} \renewcommand{\m}[1]{\mathbf{#1}} \renewcommand{\trace}[1]{\mathrm{trace}\left(#1\right)} \renewcommand{\diag}[1]{\mathrm{diag}\left(#1\right)}$ ...
0
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0answers
16 views

How is solve If A and B are similar n a matrices, then show that A and B have the same characteristic equation and therefore have the same eigenvalues

If A and B are similar n a matrices, then show that A and B have the same characteristic equation and therefore have the same eigenvalues.
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0answers
23 views

Computing the PDF of a low-rank multivariate normal

I have a question which seems simple, but I would appreciate some comments! Sometimes models involve a low-rank approximation to a covariance matrix. What confuses me is how you can compute the PDF ...
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0answers
8 views

Convergence of unit vector in dir. of $M^{k}*v$ to the principal eigenvector of M when $k\to \infty$ and M is symmetric

It is a standard fact that a square matrix $M$ of dimension $n$ has at most $n$ distinct eigenvalues, each of them a real number, and the sum of their multiplicities is exactly $n$. We will denote ...
0
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1answer
27 views

Proving Euclidian Norm squared is equivalent to transpose times matrix for x in R^n

Apologies if this has been answered already but I can't seem to find an answer that I think answers my question (or at least one I understand). Anyways the question is, ...
0
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1answer
48 views

General treatment of matrices with these properties

edit: from my exchange with Travis, I am clarifying the question (edits are in bold). Suppose we define matrices $\alpha, \beta$ with these properties: $$ \alpha=\alpha^\dagger\\ \beta=\beta^\dagger\...
1
vote
1answer
36 views

What rotations are performed to produce this output on a Tesseract?

I'm writing a program that projects a tesseract in 3D on a 2D environment and I want to reproduce the rotation of this gif but I'm having difficulty grasping what rotations before projection I need to ...
0
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0answers
27 views

Where can I find a broad set of exercises on Matrix calculus? [duplicate]

I am looking for exercises particularly on matrix differentiation - any reference textbook with theory examples is appreciated too.
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2answers
40 views

Prove of $\Vert A \Vert_\infty$ submultiplicativity

How can I prove that $\Vert AB \Vert_\infty\le\Vert A \Vert_\infty.\Vert B \Vert_\infty$ ? What I have already done: $\max_{1\le i \le n}(\sum_{j=1}^n|\sum_{k = 1}^n A_{ik}.B_{kj}|)$ $\le \max_{1\...
1
vote
2answers
43 views

Derivative of matrix exponential [on hold]

What is the derivative of $e^{(x-y)Q}$ with respect to $y$, where $x$ and $y$ are scalars and $Q$ is a transition rate matrix?
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0answers
24 views

Geometry of Solution Set and Matrix

Hello all. If wrote this system as a matrix, I would recognize it as a 3x5 matrix, but what I am not confident on is the fact that there are four variables, with constants on the other side. My gut ...
0
votes
1answer
30 views

Confused about finding non-brute force way to solve for matrix to the 2019th power

I am attempting to solve this problem, it has four parts. I solved part a (a trivial matrix problem), but the next three parts appear to be a bit confusing to me. I just would like some help getting ...
0
votes
1answer
23 views

What is the total number of distinct $m\times n$ matrices in row canonical form using only $0$s and $1$s?

Suppose that $A$ is an $m \times n$ matrix over a field $F$. What is the total number $N$ of the distinct matrices in row-reduced echelon form that are row equivalent to $A$ and that only have entires ...
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2answers
48 views

Find two vectors X, Y [on hold]

Let $$C=\begin{pmatrix} 1 & 2 & 3 \\ -1 & 1 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix}.$$ Find two vectors $X, Y \in R^3$ such that $X^TCY \neq Y^TCX$.
1
vote
1answer
19 views

Find $A(v_1+v_2)$ and $A(3v_1)$ given eigenvectors and eigenvalues

If $v_1=[-1;5]$ and $v_2=[-3;5]$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\lambda_1=-1$ and $\lambda_2=1$, find $A(v_1+v_2)$ and $A(3v_1).$ I managed to find $A,$ ...
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0answers
28 views

Can I find if the matrix is symmetric or not with missing values inside it? [on hold]

This is an Example of sample of data I am using: ...
3
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0answers
31 views

Show that the spectral norm of one matrix is smaller than the other.

Given matrices $$A = \begin{bmatrix} 0 & 0 & 0 & 1/3 \\ 0 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 1 \\ 1/3 & 1/2 & 1 & 0 \end{...
2
votes
4answers
36 views

How To Find The Unit Eigenvectors

I have the matrix $$\begin{pmatrix}3&-9\\-9&27\end{pmatrix}.$$ I found the eigenvalues of $0$ and $30.$ However, when I try to plug in $0$ for the Eigenvalues and row reduce, I get $0,0$ as my ...
0
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0answers
19 views

Confusion with the QR decomposition

I have some trouble to understand this: According to the factorization QR, Given a matrix $A\in\mathbb{R}^{n\times p}$, with $rank=min(n,p)$, there exists an orthogonal matrix $Q\in\mathbb{R}^{n\...
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vote
0answers
11 views

Eigenvalues of real part of positive definite hermitian matrix

Lets say we have the $n\times n$ positive definite hermitian matrix $\mathbf{A}$. Is there any clear relation between the Eigenvalues of $\mathbf{A}$ and the Eigenvalues of its real part $\mathbf{B}=\...
0
votes
0answers
12 views

Relationship between Cayley transform and polar decomposition

I want to (as) efficiently (as possible) numerically compute the rotation $\mathrm{R}$ in the polar decomposition of a $n\times n$ matrix of the form $\mathrm{I} + \mathrm{W}$ where $\mathrm{I}$ is ...
0
votes
1answer
27 views

Prove if a matrix A multiplied by its transpose is 0, then the matrix A is nule

if A is a matrix NxN prove that if A x A^t = matrix nule NxN so A is nule NxN I'Have tried by the summation notation but nothing came
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0answers
36 views
2
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2answers
66 views

Is $0$ an eigen value of $A$

Consider the following matrix: $A=\begin{bmatrix} 9&1&1&1&1&1&2&2\\1& 9&1&1&1&1&2&2\\1&1&9&1&1&1&2&2\\1&1&...
0
votes
1answer
17 views

Matrix notation for double sum of row and column with the same index

a biologist/ecologist here, I never took any courses in algebra (definitely missing in my education) but I am working with matrices all day. For one paper, I have to write the matrix formulation but ...
0
votes
1answer
20 views

How to show that the matrix $R^TCR$ is diagonal if $R$ is a rotation matrix related to $C$ in a specific way?

I have two matrices: $C=\begin{bmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{bmatrix}$ and $R=\begin{bmatrix}\cos\theta &-\sin\theta\\\sin\theta & \cos\theta\end{bmatrix}$. I would like to ...
0
votes
1answer
20 views

Struggling to prove that the elements of an inverse matrix satisfy a certain equation.

We have three vectors $\vec{e_1},\vec{e_2},\vec{e_3}$ that are not necessarily orthogonal or normalised, but do form a basis. We also have a matrix $G$ with elements $G_{ij}=\vec{e_i} \ . \vec{e_j}$,...
2
votes
2answers
74 views

How to compute the smallest eigenvalue efficiently? [on hold]

$A$ is a $m \times m$ symmetric PSD matrix whose top $n$ eigenvalues are equal to $1$ and whose remaining $(m-n)$ eigenvalues are zero. Here, $n \ll m$. Let $D$ be a diagonal matrix with all diagonal ...
1
vote
4answers
66 views

Let $A \in \mathbb R^{5 \times 5}$. Prove that $A^{4} + I \neq O$

Let $A \in \mathbb R^{5 \times 5}$. Prove that $$A^{4} + I \neq O$$ I understand that I have to start from $A^{4}+I= 0$ and come up with a contradiction. Is it the Cayley Hamilton I have to use?
1
vote
1answer
88 views

Prove that the determinant is greater than $1$

Let $A$ be an $n \times n$ matrix whose diagonal entries are strictly positive and off-diagonal entries are negative. The sum of the entries on each column is $1$. Prove that $\det(A) > 1$. I ...
0
votes
0answers
15 views

Can a Matrix be self inverse after performing some linear opertion on the output?

I am interested in MDS(Maximum Distance Separable) matrix which acts as self inverse when some linear operation is performed on the output. The linear operation can be swapping the values, Addition or ...