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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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\\ & & & & ...
0
votes
1answer
27 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
163 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
22 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
15 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
37 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
70 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
23 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. ...
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 ...
0
votes
1answer
30 views

Finding eigenvvalue and eigenspace

I am given a matrix $A= \bigg({} \matrix{10 & 7 \\-14 &-11} \bigg{)}$ and eigenvalue $3$. My elite mission is to find the treacherous basis for the eigenspace. I used the $(A -eI)=v$ where ...
2
votes
1answer
72 views

Two questions 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
12 views

Lagrange Method for Presenting Bilinear form as sum of squares

I have the following question in my assignment which I'm having a hard time solving. For the following bilinear form, present find a digonal form (diagonal matrix form): What I thought to do at ...
0
votes
1answer
24 views

Cauchy Schwarz Inequality for Schur product?

Let $M_{2^k}=M_2\otimes M_2\otimes\cdots \otimes M_2$ and $P_{2^k}=P_2\otimes P_2\otimes\cdots \otimes P_2$ where $M_2=\left[\begin{matrix} 1&1\\ 1&0\\ \end{matrix}\right]$ and ...
2
votes
2answers
41 views

$A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive.

Let $A$ be a $10*10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive. Then which of the following statements is always false? A. ...
0
votes
1answer
17 views

Trace distance between “weighted” Hermitian matrices

The trace norm for a matrix $\mathbf{A}$ (also known as Shatten 1-norm) is defined as follows: $\|\mathbf{A}\|_1=\operatorname{trace}[\sqrt{\mathbf{A}^*\mathbf{A}}]$. It yields a useful distance ...
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
37 views

How to show a set of vectors does not span a vector space?

Let's say I am given a $4\times4$ matrix and I am to determine whether the columns of that matrix span $\mathbb R^4$. Please tell me if I'm correct: One way to determine that is to calculate the ...
3
votes
1answer
27 views

A question about matrices such that the elements in each row add up to $1$.

Let $A$ be an invertible $10\times 10$ matrix with real entries such that the sum of each row is $1$. Then is the sum of the entries of each row of the inverse of $A$ also $1$? I created some ...
2
votes
1answer
39 views

trace inequality of positive definite matrices.

Assume $A,B \in M_n(\Bbb{R})$ are positive definite matrices, show that $$\text{Tr}(AB)\leq \text{Tr}(A)\text{Tr}(B) $$ I only prove it for $n=2$, it is straightforward calculate.but when $n \geq ...
0
votes
0answers
17 views

Converting non square matrix into square matrix [on hold]

Is it possible to convert non square matrix into square matrix? if so, how to convert?
2
votes
1answer
55 views

Number of ways to place exactly two kings in each column such that no king attacks another

A regular King in a chess board can attack all its adjacent 8 cells (vertical, horizontal or diagonal). Now you are given a $10 \times n$ chessboard, your task is to place exactly two kings in each ...
1
vote
1answer
42 views

Is $x^3$ in the null space of the transformation $p(x) \mapsto xp(x)$?

Let $h: P_3 \to P_4$ be given by $p(x) \mapsto xp(x)$. Is $x^3$ in the null space ? Or is it in the range space ? Also, I am having difficulty finding the null space and the range of this map, can ...
0
votes
2answers
15 views

Solving systems of equations using matrices by row reduction

Solve the following system for $a$, $b$, and $c$: $$\begin{pmatrix}1 & -1 & 2\\2 & -2 & 2 \\ 3 & -3 & 2\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix} = ...
1
vote
2answers
39 views

How to solve a linear system in matrix form using Laplace transform?

How to solve this linear system using Laplace transform? $$\mathbf X'(t)=\left[\begin{array}{r,r,r}-3&0&2\\1&-1&0\\-2&-1&0\end{array}\right]\mathbf X(t); ~~~~~~~~\mathbf ...
0
votes
1answer
39 views

Can a matrix have more than one inverse (Singular Value Decomposition)

Assume there's a matrix $A$ with SVD as below $$ A = U \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} ...
0
votes
1answer
44 views

fundamental matrix solution

$$ \frac{dy}{dt}=\left[\begin{array}{ccc} 5&1&1\\ 1&5&1\\ 1&1&5 \end{array}\right]$$ I need to solve this problem but my answers are still uncorrect.
1
vote
3answers
56 views

Symmetric Matrix Transformation

Here's the question, Let $T$ be the transformation of 2 by 2 real symmetric matrices defined by: \begin{bmatrix}a&b\\b&c\end{bmatrix}>>>>\begin{bmatrix}c&-b\\-b&a\end{bmatrix} ...
1
vote
1answer
53 views

Odd order n smaller than 27

I have a group $G$ that is a group of matrices of the form $$\left( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right)$$ where $a,b,c \in \Bbb Z_3$. ...
2
votes
1answer
23 views

Orthogonal decomposition of matrices

Let $M_{i,j}$ be a matrix of the standard basis of size $n>1$ (the coefficients are $1$ at position $i,j$ and $0$ elsewhere) What is the least number of orthogonal matrices required so that ...
0
votes
3answers
43 views

Approximate matrix by a rank 2 matrix using singular values

I only understand the singular value decomposition process. Do I apply it to this matrix? \begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix} What is the idea ...
0
votes
1answer
18 views

Matrix Factorization of lower triangular and unit upper triangular.

So I have a matrix. $$A =\begin{bmatrix} 8 && -3 && 2 && -1\\ -3 && 8 && 0 && 2\\ 2 && 0 && 8 && -3\\ ...