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
3answers
21 views

Finding matrix with respect to given bases

Given that A: \begin{matrix} a & b & c \\ d & e & f \\ \end{matrix} is a matrix of T : V -> W with bases G = {g1, g2, g3} and Q = {q1, q2}, respectively. Find the matrix of T ...
2
votes
1answer
29 views

If a 2x2 set of matrices are independent, do they also span M22?

I am looking for a basis in M22. So I need to get a linearly independent set that also spans the vector space. I have worked out how to tell independence, but I am stuck on the spanning requirement. ...
0
votes
1answer
11 views

Singular matrices over a commutative ring $R$, with a given adjoint matrix

First, I apologize if this is a duplicate question. I also must apologize if this has a trivial solution. This question has two parts: Let $R$ be a commutative ring with $1$, and let $F = R^n$ be a ...
0
votes
1answer
14 views

Given 2 matrices generate a reducible algebra, show they have a common eigenvector

Two matrices A, B generate an algebra.... the span of all words made with A and B... example of an element of the algebra: $A^kB^nA^m + I + B^sA^q$ etc... (exponents are all nonnegative). This algebra ...
0
votes
0answers
11 views

Matrices in the plane,polygon assignment, help. please?

a.) Given polygon P with vertices (1,5), (4,8), (8,5), (6,2) and (2,1), find the following: Find the area of P. Tip: Make sure to move COUNTERCLOCKWISE from point to point to ensure you get a ...
1
vote
2answers
17 views

Diagonalization problems (eigenvalues and vectors)

I am trying to diagonalize the following matrices: $$A = \begin{pmatrix}0 & 1\\-1 & 2\end{pmatrix}\qquad B = \begin{pmatrix}1 & 2\\-1&-1\end{pmatrix}$$ For matrix $A$, I find an ...
0
votes
3answers
42 views

Choose h and k such that the system has a solution, a unique solution and many solutions.

Im learning linear algebra, and im tasked with choosing $h$ and $k$ such that this system: $$ \begin{cases} x_1+hx_2=2\\ 4x_1+8x_2=k\\ \end{cases} $$ Has (a) no solution, (b) a unique solution, and ...
2
votes
3answers
65 views

Showing Orthogonality

How would I do this question..... I'm familiar with Gram-Schmidt and the basics but I have no idea how to do $a$ and $b$ in this question. Suppose $\{\vec x_1, \vec x_2, \vec x_3\}$ is an ...
0
votes
0answers
19 views

Properties of matrix eigenvalues

I have recently come across a $N\times N$ square matrix which has only two non-zero eigenvalues: $\lambda_1$ and $\lambda_N$. I was wondering if anyone happens to know if this is a special property ...
1
vote
1answer
25 views

Find $P$ such $P^{-1}AP = kR$.

Let $ A=\begin{bmatrix} 3 & -5 \\ 1 & -1 \\ \end{bmatrix}$ Find P such that $P^{-1}AP = k R$ where $k \neq 0$ is a scalar and R an element of $SO(2)$ = {R element of ...
0
votes
0answers
11 views

Matrices in the plane assignment, help. please?

a.) Given polygon P with vertices (1,5), (4,8), (8,5), (6,2) and (2,1), find the following: Find the area of P. Tip: Make sure to move COUNTERCLOCKWISE from point to point to ensure you get a ...
3
votes
3answers
35 views

Find the axis of rotation of a rotation matrix by $INSPECTION$ (NOT by solving $Kv=v$)

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$ by INSPECTION. This is from my other thread ...
0
votes
0answers
5 views

Median of Medians in 2D Array/Matrix

This is a bit of a mathematics problem, and a MATLAB problem. In MATLAB, if I call median(M), where M is an ...
0
votes
0answers
16 views

Unitary Farey Sequence Matrices

Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m $$ The dimension of ...
0
votes
0answers
4 views

How to calculate the combined frequencies of a DCT matrix?

Given a 2D matrix of dimensions w1,h1. I preform a DCT 2D transform on the matrix (DCT = DCT type 2). I get a 2D result matrix. This matrix has two frequency axes - x,y (which are simply the ...
1
vote
1answer
13 views

Similar Matrices Conditions:

Sorry this question was already asked but my english is not good. For matrix to be similar, does it have to have all of these properties or SOME of them? Same determinant Same Trace Same ...
0
votes
1answer
16 views

Prove that the $j$-th column of $AB$ is the product $Ab_j$

Prove that the $j$-th column of $AB$ is the product of $A$ and the $j$-th column of $B$ First of all, THIS IS NOT HOMEWORK. This was a homework. I can prove this using the fact that $e_j$ extracts ...
1
vote
1answer
27 views

Expressing a matrix in terms of subgroup generators using Magma

With Magma, it is possible to define a subgroup $H$ of a finite matrix group $G$ in terms of generators. Given a matrix $M\in G$, Magma can also determine whether $M\in H$. Presumably, if Magma ...
1
vote
1answer
23 views

Is every invertible matrix over an algebraically closed field diagonalisable?

In $\Bbb{R}$ the only invertible matrices (I can think of) that are not diagonalisable are those which stand for a rotation, but in $\Bbb{C}$ this shouldn't be a problem anymore, since rotations can ...
0
votes
0answers
7 views

$U(n)/Z(U(n))$ is isomorphic to $SU(n)/Z(SU(n))$

A problem that I have been working is withere $U(n)/Z(U(n))$ is isomorphic to $SU(n)/Z(SU(n))$. I believe the best way to approach this problem is to show that they are both Isomorphic to the same ...
0
votes
0answers
102 views

Matrix graph and irreducibility

\usepackage{multirow}!!!! How do I prove that if $A\in\mathbb C^{n\times n}$ is a matrix then it is irreducible if and only if its associated graph (defined as at Graph of a matrix) is strongly ...
0
votes
0answers
14 views

$T_1 \times T_2$ is a maximal torus?

I have been working on teaching myself matrix groups and I have come across a problem about maximal tori. If I have a torus, $T_1 \subset G_1 $ and it is the maximal torus and If I have a torus, $T_2 ...
0
votes
0answers
15 views

Transform an almost positive definite matrix to positive definite matrix.

Matrix A (n by n) is constructed as followed. $A(i,i)= -\sum^{k=n}_{k=1}A(i,k), k \neq i. $ A is symmetric. All the off-diagonal elements are negative except two elements $A(p,q)$ and $A(q, p)$: ...
5
votes
1answer
431 views

Smith normal form of a polynomial matrix.

I have the following matrix: $P(s) :=$ \begin{bmatrix} s^2 & s-1 \\ s & s^2 \end{bmatrix} How does one compute the Smith normal form of this matrix? I can't quite grasp the algorithm.
0
votes
1answer
19 views

finding unitary rotation matrix for triangularization

I have an arbitrary complex 2x2 matrix $B$ and want to find a unitary rotation $Q = \begin{bmatrix} c & -\bar{s} \\ s & \bar{c} \\ \end{bmatrix}$ with $|c|^2+|s|^2 = 1$, such that $Q^HB = ...
1
vote
0answers
20 views

Using matrices to solve systems,2 questions, pre-calc. help? [on hold]

1.) Solve for a and b. With work. [a b] [2 0] = [8 -8] .................................................................................................................................... ......[1 ...
3
votes
1answer
28 views

What is a reducible algebra?

In my matrix analysis book, a set of complex matrices is said to be an "algebra" if 1)it is a subspace, 2)whenever A and B are members, so is AB. Then it uses the terms reducible and irreducible ...
3
votes
0answers
27 views

Does the cross section of $[-1,1]^n$ on a $k$-dimensional subspace always contains a rotated image of $[-1,1]^k$?

This question is inspired by a recent bounty question, but the two questions are different and solving this one, I believe, will not lead to an answer of that bounty question. Suppose $n>k\ge1$ ...
1
vote
2answers
31 views

Trace and eigen value problem

Prove that two $n \times n$ matrices $A$ and $B$ have the same eigen values if and only if $\operatorname{trace}(A^{k}) = \operatorname{trace}(B^{k})$.
1
vote
1answer
72 views

Relative Eigenvalue Perturbation Bound deduction from Ostrowski's Theorem

I need to deduce the relative eigenvalue perturbation bound from Ostrowski's Theorem. In short i need to proove ´this statement; $\frac{|\lambda_k(SAS^*)-\lambda_k(A)|}{|\lambda_k(A)|} \leq ...
2
votes
1answer
28 views

Real or imaginary eigenvalues?

The question I have been lost in for a while is when will a matrix have either all real or complex eigenvalues? (Depending on dimensions of the matrix in question, complex and real eigenvalues may ...
1
vote
1answer
21 views

prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$.

I want to prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$. I want to use the theorem that every maximal torus of G equals $gTg^{-1}$ for some $g \in G$. But I am not ...
0
votes
1answer
29 views

Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues - Proof Strategy [Lay P397 Thm 3]

Herein, I denote the Hermitian conjugate by * (ie: $A* = \bar{A}^T) $. Let $v_i$ and $v_j$ be two eigenvectors of an Hermitian matrix H. First of all suppose that their respective eigenvalues i and j ...
0
votes
2answers
23 views

How to find the matrix of a transformation relative to standard basis?

Given $b_1=(-1,3)$ and $b_2=(1,-2)$ which make a basis for $\mathbb R^2.$ If $$ T(b_1) = 6b_1 + 7b_2 \quad\text{and}\quad T(b_2) = 3b_1 + 8b_2, $$ find the matrix of $T$ relative to the standard basis ...
1
vote
2answers
28 views

Calculating the Eigenvectors and Eigenvalues of this Matrix Polynomial

For the matrix $$ A=\begin{pmatrix} 1 & 1 & 2 \\ 0 & -2 & 0 \\ 0 & 2 & 3 \end{pmatrix} $$ How are the eigenvalues and eigenvectors of the following matrices calculated? ...
1
vote
1answer
24 views

The derivative of $x^TAx$ w.r.t $t$

Suppose $P = x^TAx$ How to find $\frac{dP}{dt}$? if $x' = Bx$ , where $B$ has the same dimension as $A$. How to find the final answer? my answer is: $$\frac{dP}{dt} = 2[(A+A^T)x]x' = ...
2
votes
3answers
47 views

Explain why $S$ is not a basis for $\mathbb{R}^3$

Explain why $S$ is not a basis for $\mathbb{R}^3$ $S=\{(1, 3, 0),(4, 1, 2),(-2, 5, -2)\}$ I set this equal to an arbitrary vector $\mathbf{x} = (x_1, x_2, x_3)$ After solving I got the matrix: ...
0
votes
0answers
14 views

Backward error for Crout factorization

Ok, can someone please tell me what is the formula for the max error in LU decomposition of Crout factorization?
1
vote
0answers
18 views

Normality of the product of a diagonal matrix and an SPD matrix?

I believe this to be true, but can't seem to prove it exactly: suppose $A$ is symmetric positive definite, and $D$ is a diagonal matrix. Then, $A$ is diagonal if $DA$ is normal for any diagonal ...
3
votes
1answer
24 views

Determinant of a matrix with symmetric positive definite block

In reviewing linear algebra for an exam, I encountered the following problem: Let $A \in \mathbb{R}^{n\times n}$ be symmetric positive definite. If $x$ is any nonzero vector, show that $$ ...
1
vote
1answer
22 views

Trace norm of Hermitian matrix

Let $A\in L(H)$ some Hermitian matrix, where $H$ is some finite dimensional Hilbertspace. I want to show $$\left\|A\right\|_{tr} = \max_{U\in U(H)}|\text{tr}(UA)| \ \ \ (*)$$ where U is unitary, and ...
1
vote
1answer
18 views

similar matrices have the same bandwidth?

If $A$ is symmetric with bandwidth $p$ then $A_+ = Q^{T} A Q$, where $Q$ is orthogonal, is orthogonally similar to $A$. How can we show/prove that $A_+$ also has bandwidth $p$ ?
0
votes
1answer
26 views

Cannot find eigenvectors

How can I find eigenvectors of the following matrix? $$ \begin{matrix} 4 & 0 \\ 0 & 1 \\ \end{matrix} $$ Systematic approach would be: 1. Finding eigenvalues ...
2
votes
2answers
102 views

Is it possible to diagonalize a singular matrix?

I have not seen anywhere written that it is impossible, but it seems impossible, so I want to check if I missed something. According to a theorem, an nxn matrix is diagonalizable if it has n ...
0
votes
1answer
20 views

Find the Axis of rotation of rotation matrix $K$ after solving $(K-I)v=0$

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$. This is from my previous thread click here ...
1
vote
1answer
37 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
1
vote
1answer
51 views

Minimzing the generalized dissimilarity measure

I am trying to solve the following problem for quite some time now, but with no progress. Here is the problem. Let $x_1....x_n$ be n samples in d-dimmensional space and let $S$ be a non ...
0
votes
2answers
20 views

How to write this b in matrix form in matlab?

Can anyone help me write this b in matrix form in matlab? I am letting n=10 for the dimension of A.
4
votes
1answer
90 views

determinant inequality, $AB=BA$, then $ \det(A^2+B^2)\ge \det(2AB) $

$A$ and $B$ are two $n\times n $ real matrices, $AB=BA$. Can we conclude that $$ \det \Big(A^2+B^2\Big)\ge \det(2AB) $$ is right? Well, the inequality is interesting. if $A,B$ are upper ...
3
votes
1answer
210 views

What does the following symbol mean? (direct sum? o-plus? — subject: matrix theory)

In this paper equation 11, the author uses a symbol that is a cross in a circle. I believe I have seen that referred to as a direct sum, but I am not completely sure what that is. $$\bigoplus$$ ...