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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0
votes
0answers
10 views

Null/Col/Row space be a line\plane through the origin?

For a $4\times3$ matrix can the $nullspace$, the c$olumn space$ and $row space$ all be a line through the origin? For a $2\times4$ matrix can the $nullspace$, the $column spac$e and $row space$ all be ...
0
votes
0answers
21 views

How can i find column of matrix corresponds to row of matrix's inverse

let $Y=X\beta$ be an equation of matrix and let $X$ be an invertible $n\times n$ matrix, $Y$ be $n \times 1$ matrix, $\beta$ be $n \times 1$ matrix. $$\begin{bmatrix} y_1 \\ y_2 \\y_3 ...
-2
votes
2answers
39 views

Nullspace, row space, column space in $m\times n$ matrices [on hold]

For a $4\times 3$ matrix can the nullspace, the column space and row space all be a line through the origin? For a $2\times 4$ matrix can the nullspace, the column space and row space all be a plane ...
0
votes
1answer
31 views

How to simplify $\det(M)=\det(A^T A)$ for rectangular $A=BC$, with square diagonal $B$ and rectangular $C$ with orthonormal columns?

Assume a real, square, symmetric, invertible $n \times n$ matrix $M$ and a real, rectangular $m \times n$ matrix $A$ such that $m \geq n$ and $M = A^T A$. Also assume that $A = B C$, where $B$ is ...
2
votes
2answers
46 views

Determinant of a 4x4 matrix with trigonometric functions

I am stuck with my homework from math. I should calcutate the determinant of a matrix: $$\begin{bmatrix} sin(x) & \sin(2x) & \cos(x) & \cos(2x)\\ cos(x) & 2\cos(2x) & ...
1
vote
1answer
29 views

Finding basis of inverse image

Let $\psi $ be a linear transformation such that$$\psi ([x_1,x_2,x_3,x_4])=[x_1+x_3+x_4, -x_2-x_4,x_1+x_2+x_3+2x_4].$$ Find basis of inverse image $\psi^{-1}(W)$ of subspace ...
1
vote
0answers
35 views

Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
0
votes
1answer
15 views

a question about general and particular solutions

We have a $3\times 6$ matrix $A$ with rank $3$ (this is all the information we have, no matrix given). Here comes the questions: What is the number of free variables in the solution to the system ...
1
vote
1answer
16 views

Odd coefficient in $M\in \mathcal{M}_n(\Bbb{Z})$ satisfies $n\le m\le n²-n+1$.

Let $M\in \mathcal{GL}_n(\Bbb{Z})$ I would like to prove that all odd coefficient of $M$ satisfies $n\le m\le n²-n+1$. In fact I don't see why $m$ is necessary bigger than $n$. I can only prove ...
0
votes
2answers
26 views

In dual numbers, what number is represented by the following matrix?

In dual numbers, what number is represented by the following matrix? \begin{pmatrix}0 & 0 \\1 & 0 \end{pmatrix}
0
votes
2answers
33 views

how to differential exponential of a matrix variable $f(X)=e^{X(t)\mathrm{d}t}$?

I have a function about a square matrix $X$ which depends also time: $f(X)=e^{X(t)}$, $t$ is time. So how to differential it about time to have $\frac{\mathrm{d}f(X)}{\mathrm{d}t}$? I know that ...
1
vote
1answer
43 views

Is this a metric on matrices?

In the set of $n$-by-$n$ reversible real matrices, decide whether $$d(A,B)=\ln (\lVert A^{-1}B\rVert\cdot\lVert B^{-1}A\rVert)$$ defines a metric and/or semi-metric. Can you please help me to solve ...
0
votes
2answers
25 views

Show that $trv=\lim_{t\to 0}\frac{\det(I+tv)-1}{t}$ for any n by n matrix

Prove that for any n by n real matrix $v\in {\mathbb R}^{n\times n}$, $trv=\lim_{t\to 0}\frac{\det(I+tv)-1}{t}$, where $t\in\mathbb R$, $I$ is the identity matirx, and $trv$ denotes the trace of ...
1
vote
1answer
20 views

Eigenvalue of altered matrix: $pI_n + qA$

As a part of an exercise I have to prove the following: Let $p,q \in \mathbb{R}$. Let $A$ be an $(n \times n)$ matrix. Let $I_n$ be the $(n \times n)$ identity matrix. If $A$ has an eigenvalue ...
8
votes
1answer
59 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
0
votes
1answer
22 views

standard matrix using a orthonormal bases

I need a small clarification. I was trying to solve the following question. If $u_1,u_2,....,u_n $ and $v_1,v_2,...,v_n$ are orthonormal bases for $\mathbb{R}^n$. construct the matrix A that ...
4
votes
1answer
41 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
24 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 ...
0
votes
1answer
38 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 ...
1
vote
0answers
13 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 ?
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 ...
28
votes
15answers
2k 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 ...
1
vote
1answer
21 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, ...
1
vote
0answers
23 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 ...
0
votes
1answer
12 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
40 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 $n\in \mathbb N$ ...
1
vote
1answer
16 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
33 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
23 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
45 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.
2
votes
1answer
241 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
46 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 ...
1
vote
1answer
18 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} & ...
0
votes
1answer
21 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?!!
0
votes
2answers
37 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
33 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
38 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 ...
1
vote
1answer
37 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 ...
0
votes
2answers
24 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
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?
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 ...
3
votes
1answer
69 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
1
vote
1answer
8 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
0answers
12 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
45 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
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 ...
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 ...
0
votes
0answers
10 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 ...
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\\ & & & & ...