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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1answer
12 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 ...
3
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0answers
18 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 ...
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2answers
18 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 ...
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2answers
27 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? ...
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0answers
20 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
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1answer
22 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' = ...
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0answers
12 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?
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3answers
46 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: ...
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1answer
23 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 ...
1
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1answer
16 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 $p$ ?
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2answers
95 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 ...
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1answer
14 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 ...
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0answers
7 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 ...
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2answers
17 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.
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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 ...
1
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1answer
19 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 ...
3
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1answer
20 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 $$ ...
4
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1answer
84 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 ...
0
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1answer
15 views

Is the adjacency matrix of every connected undirected graph irreducible? [on hold]

The adjacency matrix of every connected undirected graph is irreducible. Is this statement true?
0
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1answer
27 views

Prove a statement for the infinite matrix

We are given infinite two dimensional matrix $\{a_{i,j}\}_{i,j=1}^\infty$. And we know that matrix contain only natural values and each number appears in the matrix exactly 8 times. Task is to prove ...
0
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1answer
26 views

matrix logarithm, determination and trace maxmimzation

Let $A = A^\ast \in M_n$ be a positive definite matrix ($\lambda_i(A) > 0$). Show that $\log\det(A)-Tr(A)$ is maximized by $A = I$.
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0answers
53 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 ...
1
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1answer
50 views

Hermitian Matrix with their eigenvalues arranged in non-decreasing order

I need to formulate one property of Hermitian Matrices. It goes like this; If A, B $\in M_n$ are hermitian and their eigenvalues are arranged in non-decreasing order , then $\lambda_i(A+B)\leq ...
0
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1answer
18 views

Projection on cone of non-negative definite matrices

Ok, so if you have a real symmetric matrix $Q$ then the projection of that matrix on the cone of symmetric non-negative definite matrices $\mathcal{C}$ can be explicitly found if we do an ...
0
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1answer
31 views

How to show these two matirces are similar?

How can I show that these two block matrices are similar? $M_1 = \begin{bmatrix} AB & 0\\ B & 0\\ \end{bmatrix}$ and $M_2 = \begin{bmatrix} 0 & 0\\ B & BA\\ \end{bmatrix}$ where ...
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2answers
30 views

Find the axis of rotation from the rotation matrix.

This is a problem from the book "Mathematical Methods in the Physical Sciences" Third Edition by author Mary L. Boas. on page 129, Example 5, just in case any of you are familiar with it. So I ...
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0answers
31 views

Mapping unit sphere to ellipsoid

Consider an $N$-dimensional space. Let $M$ be a square $N\times N$ (real, but I am interested in complex case too) matrix. Are the following (hyper)ellipsoids (or degenerate hyperellipsoids)? $\{v ...
1
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2answers
18 views

orthonormal vector properties

I have noticed a matrix property that is outlined below: I have a set of n orthonormal eigenvectors that form a basis in Rn. If these vectors are combined to form an nxn matrix where each column is ...
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0answers
36 views

Matrixes and modulo of a vector

Consider an $N$-dimensional space. Consider the function $\kappa$ which maps a square $N\times N$ matrix $M$ into the scalar field $v\mapsto \lvert Mv \rvert$ (for $v$ being a vector). Is the ...
3
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2answers
162 views

Matrix multiplication question (diagonal matrices)

Suppose $AB = BA$ and $A^2+B^2 = I$, where A and B are complex matrices. My feeling is that this implies that both A and B are diagonal matrices. But I'm having trouble proving it. Appreciate any ...
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1answer
27 views

JCF of matrices $A^2$ and $B^2$

Going through a past paper and I've come across this True or False question: If $A$ and $B$ have the same Jordan Canonical Form (JCF), $A^2$ and $B^2$ have the same JCF. I thought it was true, and ...
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1answer
23 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 ...
2
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1answer
14 views

Spectral Norm of $2\times 2$ symmetric matrix

Consider a $2\times 2$ symmetric matrix, in this case, is there some closed formula for its spectral norm ? By spectral norm I mean the induced 2-norm, there is a definition here. Thanks.
1
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1answer
49 views

How to find all $3\times3$ matrices $A$ that satisfies $A^2-3A-4I = 0$? [on hold]

How to find all $3\times3$ matrices $A$ that satisfies $A^2-3A-4I = 0$?
1
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2answers
24 views

Dot product with vector and its transpose?

I'm having trouble with the statement: $$||\textbf{v}||^2=\textbf{v}\cdot\textbf{v}=\textbf{v}^T\textbf{v}$$ taking $\textbf{v}$ as a column vector in an orthogonal matrix. How can you do the dot ...
1
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0answers
22 views

Gerschgorin Theorem singularity proof

I know how to prove the Gerschgorin Theorem but how exactly would one show that there are no values of $\mu$ s.t. $\mu<0$ for which $A-\mu B$ is singular where $$ A= \begin{bmatrix} ...
4
votes
2answers
61 views

Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.

Let A,B real or complex matrixes. Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$. I demonstrated the reciprocal: $\Leftarrow )$ The two equations are ...
0
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1answer
28 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 ...
1
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2answers
51 views

For which $a$ is a matrix $A$ diagonalizable?

Say I have a matrix $A_a$ with $$A_a:= \left(\begin{array}{c} 2 & a+1 & 0 \\ -a & -3a & -a \\ a & 3a+2 & a+2 \end{array}\right)$$ I was wondering if there was an ...
0
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1answer
21 views

Determinant of a square matrix with main diagonal of zeros?

How can I show that the determinant of a square matrix A of dimension NxN with all elements equal to $-\delta$ except the main diagonal composed by zeros, is equal to $-(N-1)\times \delta^N$?
3
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0answers
35 views

Matrix similarity problem (complex, real)

I'm trying to solve this problem: Given complex matrices A and B, prove there's a nonsingular real matrix Q such that $A=QBQ^{-1}$, if and only if there's a nonsingular complex matrix S such that ...
1
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1answer
40 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 ...
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0answers
9 views

Asymmetric n*n positive matrix factorization methods?

It's better that original space does not rotate (SVD rotates the axis). I think cholesky decomposition is nearly possible but it requires the matrix should be symmetric. Some suggestions?
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2answers
28 views

Significance of an eigenvector being equal to a unit vector?

I was reading ahead in my math book when I came across a matrix denoted as A = $\begin{bmatrix} 1 & -1 & 0\\ 2 & -2 & 0\\ 6 & 0 & -2\\ \end{bmatrix}$. I then found the ...
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1answer
38 views

Find Determinant of A

I've tried creating a triangular matrix, tried row reducing but can't figure it out as I keep on having c-unknown in my answer. How would I do this?
2
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1answer
36 views

How to find the inverse of this particular symmetric matrix

Basically, I have a $n \times n$ symmetric matrix, which looks like this: $$ \begin{bmatrix} 1 & \alpha & \cdots & \alpha \\ \alpha & 1 & \cdots &\alpha \\ \vdots &\vdots ...
1
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1answer
28 views

The inverse of a matrix (main diagonal $2$, left and right of it $-1$)

I want to find inverse matrix of the ...
0
votes
2answers
101 views

Proof: $Ax=x$ for all $x$ implies $A=I$ [on hold]

Let $A$ be a square matrix of order $n$ and let $x$ be an $n$-vector. Prove that if $Ax=x$ for all $x$, then $A=I$. Thanks in advance
1
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4answers
80 views

Prove the matrix satisfies the equation $A^2 -4A-5I=0$ [on hold]

How to prove that $$ A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} $$ satisfies the equation $A^2 -4A-5I=0$?
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0answers
31 views

Markov Transition Matrix

I have some data, shown below. How do I construct a transition matrix, for Markov Chain ? I need the formula to calculate observation data into transition matrix. Thanks! Accumulative ...