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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-2
votes
0answers
23 views

property of a matrix with row sum equal to one

Let $A$ be an $10 \times 10$ matrix with row sum equal to $1$ for each row. Then 1) $A^{-1}$ has row sum equal to $1$ for each row 2) $A^{-1}$ has column sum equal to $1$ for each column How to ...
0
votes
1answer
14 views

Algorithm for the Hill cipher (finding the inverse of the determinant of a $2 \times 2$ matrix modulo $26$)

I have a good understanding of how to do the Hill cipher on paper but putting it into program form is somewhat of a problem. Finding the the determinant is the thing I'm having problem with. On ...
0
votes
1answer
29 views

Find a minimal spanning set of a set of matrices

I'm supposed to find a minimal spanning set of $W = \{A \in M_n(\mathbb{R}) | \operatorname{Tr}(A) = 0\}$ First of all, what is a minimal spanning set? I can't find the term anywhere in the notes my ...
4
votes
1answer
34 views

Is every symmetric matrix diagonalizable?

I know that Hermitian matrices are always diagonalizable and real symmetric matrices are real Hermitian matrices and therefore diagonalizable. But, it is always not the case that a symmetric matrix ...
0
votes
0answers
20 views

Is the ratio trace problem convex?

I have a ratio trace problem described as follows: $\arg\max_{w} trace((w^tAw)*inv(w^tBw))$, where A and B are full rank matrices. This problem can be solved via generalized eigenvalue problem. ...
0
votes
0answers
14 views

$\Phi(t)=P(t)e^{tR}$ as a fundamental set for $x''(t)=\sin(t)x'(t)$

Problem. Find $2\times2$ matrices $R$ and $P(t)$ such that $R$ is constant, $P(t)$ is periodic, and $\Phi(t)=P(t)e^{tR}$ is a fundamental set of solutions for $x''(t)=\sin(t)x'(t)$. $ $ Attempt at ...
12
votes
12answers
561 views

What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
3
votes
1answer
22 views

Connected components $0-1$ matrices

Let $M$ be a $0-1$ matrix. Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...
1
vote
0answers
12 views

Basis for span and transpose of span of matrix?

Does the rows of the RREF of the transpose of the span of a matrix yield a basis of a matrix ? Can a basis also be composed of the rows RREF of the span of a matrix ?
1
vote
0answers
17 views

Give the following linear transformation find values of parameter

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4 $ When this transformation is epimorphic i.e. what ...
1
vote
1answer
19 views

Examples of Unitary Matrices with coefficients all having the same amplitude

I am looking for examples of unitary matrices like this one $$A = \frac{1}{\sqrt{2}}\left( \begin{array}{rr} 1 & 1 \\ 1 &-1 \end{array} \right)$$ where each coefficient has the same amplitude, ...
0
votes
1answer
299 views

All principal minors are equal to zero

Let's assume that all principal minors of symetric square matrix $A$ ($n\times n$) are equal to zero, then what definiteness does this matrix have? It's obvious that it's semidefinite, of course. But ...
0
votes
1answer
11 views

Extensions on matrix factorization [on hold]

The traditionl matrix decomposition [1] has the following general form: $ M = A^\top B $ or $ M = A^\top U B $ Is there any decomposition method (or smart trick using the existing methods) to get ...
1
vote
2answers
37 views

let A be a $2\times 2$ matrix . Then the smallest number $n\in \mathbb N$ such that $A^n=I$ is

let A be a $2\times 2$ matrix $\begin{pmatrix} \sin \frac \pi {18} & -\sin \frac {4\pi} {9}\\ \sin \frac {4\pi} {9}&\sin \frac \pi {18}\end{pmatrix}$. Then the smallest number $n\in \mathbb N$ ...
15
votes
9answers
958 views

Why do determinants have their particular form?

I know that for a matrix $A$, if $\det(A)=0$ then the matrix does not have an inverse, and hence the associated system of equations does not have a unique solution. However, why do the determinant ...
-2
votes
0answers
45 views

$A$ be a $10\times 10$ matrix over $\mathbb R$ such that sum of each row is $1$. [on hold]

Let $A$ be an invertible $10\times10$ matrix over $\mathbb R$ such that sum of each row is $1.$ Then which option is correct? A. The sum of the entries of each row of the inverse of $A$ is ...
1
vote
1answer
15 views

Setting corresponding entries in a matrix

I've recently read "Matrix Inversion and the Great Injustice", a rather humorous article of a student venting his frustrations due to feeling as if he has been graded unfairly. I follow everything so ...
2
votes
1answer
90 views

How find this invertible matrix $C=\begin{bmatrix} A&B\\ B^T&0 \end{bmatrix}$

let matrix $A_{n\times n}$,and $\det(A)>0$, and the matrix $B_{n\times m}$,and such $rank(B)=m$,and let $$C=\begin{bmatrix} A&B\\ B^T&0 \end{bmatrix}$$ Find this Invertible matrix ...
2
votes
1answer
28 views

Determinantal inequality for block matrices: if $A=(B,C)$ is a square matrix, then $|A|^2\le |B^TB|\cdot |C^TC|$

Suppose $A=(B,C)$ is a $n\times n$ matrix, $B$ is a $n\times s$ matrix, $C$ is a $n\times (n-s)$ matrix. Show that $|A|^2\leq |B^TB|\cdot |C^TC|$. If $A$ is singular, then it is obvious. If $A$ is ...
0
votes
0answers
22 views

What is the proof of this? (Matrices, Pivot)

I have a matrix : $A$ I pivoted $A$ with a pivot element $(p)$ and I get this matrix: $B$ What is the proof of this equation? $|A|$ = $\frac{1}{p}. |B|$
0
votes
2answers
40 views

Permutation matrix 56x56

I want to find all possible permutation matrices for an identity matrix. I need it at a 56x56 level. An explanation at a lower level would also help.
-1
votes
2answers
84 views

Are irreducible, positiv-definite Markov chains aperiodic?

If $M$ is the transition matrix of a discrete Markov chain, and $M$ is both irreducible, symmetric and positiv-definite, is the resulting Markov chain necessarily aperiodic? In my intuition, ...
2
votes
1answer
232 views

How to find the exponential of a nilpotent matrix?

I want to find the exponential $e^{tA}$, where $A=\left( \begin{array}{ c c } 0 & 1 & 2 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{array} \right).$ I know that its ...
2
votes
1answer
48 views

Solving equation with integer matrices as unknowns

I am currently working on a problem, where I need to know for square integer matrices, $A$ and $B$, whether or not there exists square integer matrices, $X$ and $Y$, such that $X(A-I)Y=B-I$, where $I$ ...
1
vote
1answer
17 views

Transforming a square matrix A into B

Let's say I have $A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix}$ and $B= \begin{bmatrix} b_{11} & ...
2
votes
1answer
45 views

Can anyone check these true and false statements about linear algebra?

For any square matrix $A$, the image of $A^7$ is contained in the image of $A$ I think this question is asking If $A^7x=b$, then $b$ must be in $A$ with some vector $y$ such that $Ay=b$. It Seems ...
0
votes
1answer
19 views

find the eigenvalue of $A^m$

Let $$A = \pmatrix{7&9\\-3&-5},$$ it is a $2\times 2$ matrix. For every integer $m$, find all eigenvalues of $A^m$, and bases for the corresponding eigenspaces How to get it?!!
4
votes
3answers
435 views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
0
votes
2answers
34 views

How to find eigenvalues of this matrix

How to find eigenvalues of this matrix: $\left( \begin{array}{ c c } 2 & 0 & 0 \\ 0 & 2 & 4 \\ 0 & -1 & 2 \end{array} \right) $ ATTEMPT: $2-λ [(2-λ)(2-λ) ...
1
vote
0answers
31 views

Name for matrices with $a_{ij} + a_{ji} = 1$?

Do you know of any commonly used name for square matrices $A$ having the property that $$ a_{ij} + a_{ji} = 1$$ for all $i,j \in \{1,\dots, n\}$, where $n$ is the dimension of $A$?
1
vote
1answer
28 views

About diagonalization

"Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
0
votes
2answers
23 views

$A$ has full rank iff $A^H A$ is invertible

Let $A \in \mathbb{K}^{m,n}$ be a matrix. How to show that $\text{rank}(A) = n$ if and only if the matrix $A^HA$ is invertible?
1
vote
1answer
34 views

Square matrices A and B commute if and only if they share the same generalized eigenspace.

I found a couple of proofs for this theorem but only for the case when A and B are diagonalizable, thus the eigenspace that they share is not the generalized one. Im looking for the proof (or ...
1
vote
1answer
6 views

Joint spectral radius of $\sigma( \mathcal A)$ and $\rho(A) < 1 \forall A \in \mathcal A$

Given $\mathcal A \subset R^{n \times n}$. The joint spectral radius is by: $$\sigma( \mathcal A) = \limsup_{m \rightarrow \infty}\sup_{A \in \mathcal A^m}\rho(A),$$ where $\rho$ is the normal ...
0
votes
1answer
24 views

Every idempotent matrix is diagonalizable.

Show that every idempotent matrix is diagonalizable. What can you say if $A$ is tripotent ($A^3=A?$) What if $A^k=A?$ The first two cases is obvious since we can find the minimal polynomial to be ...
1
vote
1answer
24 views

Complex matrix operations question

If we have $4$ Real matrices $A,B,C,D$, is it possible to show that $(A+Bi)(C+Di) = E+Fi$ in $3$ nxn matrix multiplications?
2
votes
2answers
884 views

Q: The determinant of a matrix $A \in \mathbb{R}^{n \times n}$?

I really struggle with this problem, how do you calculate the determinant of matrix $A \in \mathbb{R}^{n \times n}$, whose expression is $$ \begin{pmatrix} 2 & 1& ...& 1\\ 1& ...
1
vote
1answer
15 views

Compute $\left(a_{i}A+B\right)^{-1},\qquad i=1,\ldots,N$ efficiently?

I need to compute the inverse matrix: $$(a_i A+B)^{-1}, \qquad i=1,\ldots,N$$ where $N$ is a large number. $A$ and $B$ are general $M\times M$ matrices independent of $i$. The only thing that ...
0
votes
1answer
28 views

Finding representation matrix of $M$*$(K_5)$ above $\mathbb F_2$

Finding representation matrix of $M$*$(K_5)$ above $\mathbb F_2$. $M$*$(K_5)$ is the dual matroid representing the graph $K_5$, that is, a complete graph with 5 vertices. How do I solve this? ...
3
votes
1answer
68 views

Why is this change of basis useful?

In my textbook there is a theorem which states Let $A$ be a real $2\times 2$ matrix with complex eigenvalues $\lambda =a\pm bi$ (where $(b\ne 0)$. If $\mathbf x$ is an eigenvector of $A$ ...
-2
votes
0answers
22 views

How do you create a three dimensional matrix from a two dimensional matrix? [on hold]

I have a $2D$ logical matrix $(1765x2688)$ so I want to create a $3D$ matrix from it $(1765x2688x90)$ with same values in each level of $3^{rd}$ dimension. thanks a lot moradi
0
votes
0answers
11 views

Location and perturbation of eigenvalues

This is a problem from Horn and Johnson's Matrix Analysis. I'm having trouble showing the bolded parts in the following paragraphs. In fact, I don't really understand what the sentences mean. I would ...
2
votes
2answers
41 views

Square root of these $2\times2$ matrices

I am to find the matrix square root of $A$ from the following formula: $R=S^{-1}\sqrt{\Lambda S}$ and explain why there is no real matrix square root of $B$. I am stuck on A as the following ...
2
votes
0answers
11 views

Column and row vectors (spinors) in Landau-Lifshitz vol.IV Theoretical Physics

I am getting confused by the notation the authors of this book since they define: $$ \bar{\psi}\equiv \psi^\ast \gamma^0 $$ where (I suppose) $^\ast$ means complex conjugate and $\gamma^0$ is one of ...
2
votes
1answer
27 views

$A_i \sim B_i \implies \text{Diag}(A_1 \ldots A_n) \sim \text{Diag}(B_1\ldots B_n) $ [on hold]

How do I prove that: $A_i \sim B_i \implies \text{Diag}(A_1 \ldots A_n) \sim \text{Diag}(B_1\ldots B_n) $ Notation: $A\sim B$ meaning is $A$ is similar to $B$. Also, $A_i, B_i$ are square matrices ...
1
vote
2answers
160 views

If A is normal, then the nullspace of A is the nullspace of A*

Suppose $A$ is a normal matrix. Prove that $x$ is in the nullspace of $A$ if and only if $x$ is in the nullspace of $A^{*}$. This isn't a homework problem. It was on a test I took recently, and I'd ...
0
votes
1answer
25 views

Eigenvalue inequalities for Hermitian matrices

This is a problem from Horn and Johnson's Matrix Analysis. I've tried to follow the problem but I can't find a way to lead to the conclusion the problem is suggesting. Any solutions, hints, or ...
0
votes
1answer
26 views

what is the convex hull of such a matrix cone?

A matrix cone is in the following form: $M: = \begin{pmatrix} 1 \\ x\end{pmatrix}\begin{pmatrix}1 & x^T\end{pmatrix}$ where $x\in F$ , let $F = \{x: x\in [l,u]^n\}$ How to express the convex ...
0
votes
0answers
36 views

Rotation matrix around one coordinate in N dimensions

Probably a very simple question: Given the standard Cartesian coordinate matrix, $$\begin{pmatrix}1 & \\ & 1 & \\ & & 1\\ & & & 1\\ & & & & ...
0
votes
0answers
9 views

norm bounded Convolution in matrix space

There is a stable matrix $A$ with eigen values in unit circle,for discrete time system : $x(k+1)=Ax(k)+f(k)$ can we prove: $||\Sigma_{j=0,..,k} A^{k-j}f(j)||_2<= ||f(k)||_2/{(1-A_{max})} $ where ...