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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1
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0answers
9 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
19 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 ...
6
votes
10answers
149 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
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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, ...
1
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0answers
16 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
10 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
36 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
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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
25 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? $|A|$ = $\frac{1}{p}. |B|$
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2answers
39 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
231 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
42 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
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1answer
16 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
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?!!
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
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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 ...
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 ...
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
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
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 $A$ ...
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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
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
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
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
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 ...
0
votes
0answers
35 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
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
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? ...
6
votes
2answers
65 views

Linear maps for $\Bbb{R}^n$ to $\Bbb{R}^m$?

This question is related to: What is $\Bbb{R}^n$? The basis of a matrix representation I am still confused about the topics in these questions and am going to ask another question that will ...
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
0answers
21 views

A problem on Gersgorin cirle passing through the eigenvalue of an absolute matrix

I'm having trouble solving the following problem. I think I need to show that the matrix $D^{-1}|A|D$ has property SC, but I can't come up with a way to show it. I would really appreciate any ...
0
votes
1answer
14 views

A problem about a theorem on irreducible matrix

I'm stuck on a problem where I need to find a counterexample. I'm not sure how to come up with a reducible matrix to show that it doesn't satisfy the result of the following corollary. Any solutions, ...
0
votes
1answer
31 views

Non-simplifiable permutation matrices

The permutation matrices for 2 and 3 dimensions look like this: 2-dimensional: $$\quad M_1^{2d}=\left(\begin{matrix}1 &0\\0 &1\end{matrix}\right), M_2^{2d}=\left(\begin{matrix}0 &1\\1 ...
0
votes
1answer
35 views

What values must $\alpha$ be so that $F$ is an isomorphic linear transformation? (Bijective)

Let $F:P_2\to P_2$ where $P_2$ is a polynomial vector space with max grade of 2. $$[F]_B= \begin{pmatrix} \alpha & -1 & -1 \\ -6 & \alpha +1 & 0 \\ ...
1
vote
1answer
32 views

What's the name of these matrices

I'm searching for the name of these sets of matrices: 2-dimensional: $$\quad M_1^{2d}=\left(\begin{matrix}1 &0\\0 &1\end{matrix}\right), M_2^{2d}=\left(\begin{matrix}0 &1\\1 ...
5
votes
4answers
66 views

Whether a $2 \times 2$ matrix of rank $1$ has a zero eigenvalue

"Does $A = \begin{bmatrix}1&2\\2&4\end{bmatrix}$ have a zero eigenvalue?" Well, it would be a funny question to ask if the asker didn't state that he wants us to explain without computing the ...
2
votes
1answer
48 views

Is there an error in my matrix proofs (Also: potato quality jpeg errors present)

Disclaimer: The jpg quality of the problem is terrible, ALL SUPERSCRIPT IN BLOCKQUOTES CAN BE INNACURATE. $A\in \Bbb R^{n\times n} $ is symmetric $B\in \Bbb R^{n\times h}$ ...
0
votes
1answer
16 views

Suppose that $A \in M_n$ is strictly diagonally dominant. Show that $|a_{kk}|$ $\lt C_k'$, for at least one value of $k$

Suppose that $A \in M_n$ is strictly diagonally dominant. Show that $|a_{kk}|$$\gt C_k'$, for at least one value of $k=1,\dots, n$, where $C_k'$ denotes $A$'s deleted absolute column sums ($a_{kk}$ is ...
2
votes
1answer
18 views

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. If $A$ is real, show that every eigenvalue of $A$ is real.

Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. (a) If $A$ is real, show that every eigenvalue of $A$ is real. (b) If $A \in M_n$ has real main diagonal entries and its ...
0
votes
1answer
20 views

Proving Two Hermitian Matrices have the same eigenvectors

I am currently stuck on the following proof. Suppose that a (n by n) unitary matrix U can be written as U=M+iN where M and N are Hermitian matrices. Now assuming that M and N have n distinct ...
2
votes
1answer
15 views

Show that the intersection taken over the Gersgorin discs of all similar matrices of $A$ $=$ $\sigma (A)$

Show that $\bigcap_S G(S^{-1}AS)$ $=$ $\sigma (A)$; the intersection is taken over all nonsingular $S$, and $\sigma (A)$ is the spectrum of $A$. I'm lost as how to even begin to prove this fact. Any ...
0
votes
0answers
13 views

How to determine roll-pitch-yaw parameters from homogeneous transformation matrix

We have a transformation matrix $T = \begin{pmatrix} cos(\theta_1) & sin(\theta_1) & 0 & l_1 cos(\theta_1) \\ sin(\theta_1) & -cos(\theta_1) & 0 & l_1 cos(\theta_1) \\ 0 ...
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 ...
0
votes
0answers
13 views

Is the ratio trace problem convex?

I have a ratio trace problem described as follows: $\arg\max_{w} trace(w^tAw/w^tBw)$, where A and B are full rank matrices. This problem can be solved via generalized eigenvalue problem. My ...
2
votes
1answer
18 views

Relationship between similarity and having the same minimal polynomial

Let $A$, $B$ $\in M_3$ be nilpotent, where $M_3$ is the set of all complex 3by3 matrices. Show that $A$ and $B$ are similar if and only if $A$ and $B$ have the same minimal polynomial. Is this true in ...