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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.

2
votes
2answers
37 views

Let $A$ be a $n\times n$ matrix such that $a_{ij}=ij$,find the eigenvalues of the matrix $A$.

Let $A$ be a $n\times n$ matrix such that $a_{ij}=ij$. I want to find the eigenvalues of the matrix $A$. Efforts: $n=2$, I found that eigenvalues are equal to $0,5$. For $n=3$, I found eigenvalues ...
3
votes
3answers
26 views

Give a 2*2 block matrix $M = \begin{bmatrix}A&B\\0&C \end{bmatrix}$ and find a formula for $M^{-1}$ in terms of $A$, $B$, and $C$

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section2.5 Problem 29, I was puzzled at solving it. Here is the problem description: Give a 2*2 ...
1
vote
1answer
53 views

How to calcute a determinant of this matrix?

\begin{vmatrix} -t & 1 & 0 & 0& \ldots & 0 & 0 &0\\ n & -t & 2 & 0& \ldots & 0 & 0 &0\\ 0 & n-1& -t & 3&\ldots & 0 &...
0
votes
0answers
9 views

How to vectorize/matricize multivariate Gaussian PDF for more efficient computation?

Context: I was recently implementing (in Python) the Expectation-Maximization (EM) algorithm for Gaussian mixture models, and part of that process involves computing the Gaussian PDF for various ...
-3
votes
1answer
27 views

A=-A^T, prove the diagonal entries equal 0

Let A be a square matrix satisfying $A$=-$A^T$ Prove that the diagonal entries are all zero I was able to solve this question in the case of a specific 2x2 matrix,As shown here But I am unsure of how ...
0
votes
0answers
11 views

“Working with Representations” chapter of “Interactive Computer Graphics A Top-Down Approach” left me confused

So basically the author in that chapter says that Given two representations a and b of a point or vector in homogeneous coordinates and matrix M that is a matrix representation of the change of ...
0
votes
1answer
33 views

For a $2 \times 2$ matrix $C$, prove that $C = AB - BA$ can only be found if and only if $C_{11} + C_{22} = 0$.

This question appears in Hoffman-Kunze Linear Algebra (exercise 1.5, question 8). The book has only introduced linear equations, elementary row operations, row-reduced echolon matrices, and matrix ...
-1
votes
0answers
17 views

what is the partial derivative of $tr[\rho\log(\sum_xp_x\rho_x)]$ with respect to $p_x$? [on hold]

anyone knows the how to get $\frac{\partial}{\partial p_x}tr[\rho\log(\sum_xp_x\rho_x)]$? The "tr" means trace and $\rho,\rho_x$ are positive semi-definite matrices.
2
votes
0answers
20 views

Use bilinear form to show that all symmetric matrixes are diagonalizable

Let $B_1,B_2$ be bases for a vector space $V$. Let $g$ be a symmetric bilinear form. I have shown that if $A$ is the matrix representation of $g$ using basis $B_1$ and $B$ is the matrix representation ...
0
votes
1answer
25 views

Intuition underlying a linear algebra result

RESULT For any $m\times n$ matrix $\textbf{A}$ and $m\times p$ matrix $\textbf{B}$, $\mathcal{C}(B)\subset\mathcal{C}(A)$ if and only if there exists an $n\times p$ matrix $\textbf{F}$ such that $\...
1
vote
0answers
19 views

Eigenvalues of the product of one diagonal and one regular matrix

I'm reading a paper that makes the following claim that I don't understand. Let $\Sigma, K\in\mathbb{R}^{n,n}$, where $\Sigma$ is a diagonal matrix with only non-negative real entries. Then the claim ...
0
votes
1answer
43 views

Is there always a complete, orthogonal set of unitary matrices?

The set of size-$n$ unitary matrices span $\Bbb C^{n \times n}$ (this can be proven nicely using polar decomposition). If we select a maximal linear subset of unitary matrices, then we have a basis ...
0
votes
0answers
30 views

Real valued vector representation of Hermitian matrix

One can write a symmetric matrix $M \in \{\mathbb{R}, \mathbb{C}\}^{n \times n}$ in a half-vectorized representation, e.g., $$ M = \begin{bmatrix} a & b\\ b & c \end{bmatrix} \rightarrow \vec ...
0
votes
1answer
21 views

$Ker(A)=\{0\} \Leftrightarrow rank(A)=n$

Let $A: n\times n$, I want to prove that if $N(A)=\{0\}$ if, and only if, $A$ has $n$ linearly independent columns. Important: You cannot use $dim(N(A)) + dim(R(A)) = n$. I thought of doing so, as $...
0
votes
1answer
32 views

Let $A\in M_3(\mathbb{Q})$ so that $A^8=I_3$. Prove that $A^4=I_3$

Let $A\in M_3(\mathbb{Q})$ so that $A^8=I_3$. Prove that $A^4=I_3$. I tried to look at the solution provided by my book, but I can't understand it. They say that $A$'s minimal polynomial divides the ...
0
votes
0answers
33 views

Dimension and Base of $(E_{12} + E_{21} + E_{23} + E_{32})A = 0$ Matrix.

You have a set: $W = {A ∈ Mn(R) | (E_{12} + E_{21} + E_{23} + E_{32})A = 0}$. You need to find out if this set is subspace of $M_n(\mathbb{R})$. I proofed it that way: $(E_{12} + E_{21} + E_{23} + ...
-2
votes
1answer
21 views

Index (x_i) with two rows in latex [on hold]

I have following problem in latex formatting: I want to write for a matrix : $ (a_{ij})_{\underset{j=1,\ldots , n}{i=1,\ldots ,m}} $ but I want the $i=\ldots$ and $j=\ldots$ to be the same size... ...
0
votes
0answers
15 views

If I have a matrix in the form $Ax=B$, why must I have $\begin{bmatrix}i\\j\end{bmatrix}=β\begin{bmatrix}-2\\1\end{bmatrix}?$

I have inserted numbers in place of i and j, and have successfully deduced the value of β as a coefficient. However, i am unsure why we must have that equation if we know $A\begin{bmatrix}i\\j\end{...
1
vote
0answers
41 views

Rewrite a condition on a $3\times3$ matrix

Consider the $3\times 3$ matrix $$ A\equiv \begin{pmatrix} \mu_1-\mu_1'& \mu_1-\mu_1'-c & \mu_1-\mu_1'-c-d\\ \mu_1+a-\mu_1'& \mu_1+a-\mu_1'-c & \mu_1+a-\mu_1'-c-d\\ \mu_1+a+b-\mu_1'&...
0
votes
1answer
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 ...
1
vote
2answers
12 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 ...
0
votes
0answers
18 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 (...
0
votes
0answers
10 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
votes
0answers
59 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
21 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, ...
1
vote
1answer
22 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 (...
0
votes
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
votes
1answer
31 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 $...
0
votes
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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
13 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
votes
0answers
26 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 ...
0
votes
0answers
30 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_{ \| ...
0
votes
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
votes
0answers
53 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
48 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
votes
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
votes
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
votes
0answers
15 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.
0
votes
0answers
22 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 ...
0
votes
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
votes
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
votes
1answer
39 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
35 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
votes
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.
0
votes
2answers
39 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
40 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?
0
votes
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 ...