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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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35 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\...
2
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2answers
30 views

Number of matrices , such that sum in each row and each column is 0

To find number of 4 cross 4 matrices, such that each element is 1 or -1. Also sum of elements in each row and each column should be zero. I am able to think that each row and each column shall ...
0
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0answers
14 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
40 views

How to break the quadratic form $x^TABx + x^TB^TAx$?

Let $Q = AB$, where $A \in \mathbb{R}^{n \times n}$ is symmetric and positive definite, and $B \in \mathbb{R}^{n \times n}$ is square but not necessarily symmetric nor positive definite. Then claim: $...
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1answer
12 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 ...
2
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0answers
28 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
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1answer
17 views

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

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 ...
4
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2answers
105 views

Matrices and Combinatorics are a bad combination.

Let $\scr A$ be the set of all $n\times n$ symmetric matrices all of whose entries are either $0$ or $1$ and such that if $n$ is even, $n^2/2$ of these entries are $1$ and $n^2/2$ of them are $0$, and ...
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0answers
11 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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1answer
39 views

The matrix A=[-2 2 1 3 ] is invertible with A^(-1)=1/8 [-3 2 1 2 ].

The matrix $A=\begin{pmatrix} -2 & 2 \\ 1 & 3 \end{pmatrix}$ is invertible with $A^{-1}=\frac{1}{8} \begin{pmatrix}-3 & 2\\ 1 & 2 \end{pmatrix}$. (TRUE/FALSE)? In my opinion the ...
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11 views
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0answers
37 views

Changing rotation center

Things that we have: 2 dimensions, a object with it's coordinates (object P1), it's rotation center (pivot) C1. After that lets rotate it at pivot C1 by known angle A. Now let's move that pivot by ...
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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 ...
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0answers
12 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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2answers
62 views

How to compute the smallest eigenvalue efficiently?

$A$ 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 ...
0
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1answer
26 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, ...
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0answers
23 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 ...
20
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1answer
427 views

When do two matrices have the same exponential?

Let $A$ and $B$ be $n\times n$ hermitean matrices. When do we have $e^{iA}=e^{iB}$? Can we somehow classify those pairs of matrices that have the same exponential? Here are some observations that I ...
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2answers
69 views

Is commutation preserved for $e^A$?

Let $A, B \in \mathbb C^{n \times n}$. Suppose $A$ commutes with $B$. Does $e^A$ necessarily commute with $B$? If that is the case, consider $$S = A + B \exp(-S)$$ where $S$ is an $n \times n$ matrix....
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2answers
35 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\...
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0answers
25 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
38 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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2answers
43 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 $X^TCY \neq Y^TCX$.
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2answers
64 views

What is the probability of the matrix being Singular

Consider the set of all boolean square matrices of order $3 \times 3$ as shown below where a,b,c,d,e,f can be either 0 or 1. $\begin{bmatrix} a&b&c\\ 0&d&e\\ 0&0&f ...
13
votes
1answer
246 views

Simultaneously vanishing quadratic forms

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
0
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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 ...
0
votes
1answer
28 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 ...
1
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1answer
151 views

How to solve this quadratic equation in matrix $X$?

Given $\mathbf{A}$ and $\mathbf{B}$ two $m \times n$ real matrices, is there a closed form for the matrix equation \begin{equation} \|\mathbf{X}\|^{2}_{F} - 2 \cdot \mbox{trace} (\mathbf{X}^T\...
0
votes
0answers
21 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 ...
1
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0answers
42 views

On solving quadratic matrix equations

I am in the 12th grade and I am not that good with algebra. Is there an easy way to solve quadratic matrix equations of the form $X^2 = A$ (or even $X^3 = A$) over $M_{nxn} (\mathbb C)$, rather than ...
1
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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
26 views
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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 ...
3
votes
0answers
25 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{...
1
vote
1answer
47 views

Proof - Two entries on main diagonal will be the same for all powers

Prove that in any power of the matrix $$\begin{bmatrix} 2&1&1\\0&2&1\\1&1&1\end{bmatrix}$$ two entries on the main diagonal will be the same. I was able to prove this by ...
2
votes
2answers
64 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&...
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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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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}=\...
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votes
2answers
92 views

Proof that in any power of a matrix, two entries on the main diagonal will be the same

Say you are given a 3x3 matrix, where two elements on the diagonal of the matrix are the same, how would you go about proving that with any power of the matrix, two entries on the diagonal will still ...
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0answers
8 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 ...
5
votes
2answers
502 views

Let $A$ be real symmetric $n\times n$ matrix whose only eigenvalues are 0 and 1. Pick out the true statements.

Let $A$ be real symmetric $n\times n$ matrix whose only eigenvalues are $0$ and $1$. Let the dimension of the null space of $A-I$ be $m$. Pick out the true statements. The characteristic ...
0
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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 ...
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4answers
57 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?
0
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1answer
26 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
123 views

Creating a Generator Matrix in Standard Partition Form using a Generator Set for Linear Binary Code

Take for example, I have a generator set that is linear binary code... $\langle0011101_2, 0101011_2, 1000111_2\rangle$ And I would transform this to a Generator Matrix... (No full row or column of '...
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3answers
74 views

Why is operator norm defined the way it is?

Is there an intuition for the infimum definition of ${\| A \|}_{\mathrm{op}}$ without using a different, equivalent definition? I am referring to the definition, given an operator $A: W \rightarrow V$...
0
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1answer
19 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 $G_{ij}=\vec{e_i} \ . \vec{e_j}$,...
1
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1answer
46 views

How to form Matrix pairs of $m$ friends and to find common friend from all possible pairs.

The below is a problem given in entrance exam. Problem: A golf club has $m$ members with serial numbers $1, 2 . . . , m$. If members with serial numbers $i$ and $j$ are friends, then $A(i, j) = A(j, ...
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3answers
84 views
+300

Minimum Elementary row/column transformations to find Inverse of given Matric

While working out some elementary transformation to find Inverse of matrix, it get in my mind, what is Minimum such transformations needed to find Inverse.