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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2answers
9 views

How to check whether the given matrix is a sub matrix of another matrix?

I basically want to know whether there is an easy and straight-forward method for checking and if possible, constructing a matrix which avoids a particular matrix. For example, consider the given ...
0
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1answer
19 views

Prove that the given block matrix is positive semi-definite

How do I show $M = \begin{bmatrix} A & B \\ B^T & C \end{bmatrix} \succeq 0$ i.e. $M$ is positive semi-definite (PSD) given that $A$ is PSD and for some $\Lambda = \text{diag}(\lambda_1, ...
0
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2answers
41 views

What really makes a matrix diagonalizable?

I consider linear algebra a fascinating tool, that I believe is worth mastering. So, choose a field $\mathbb{K}$, and we start doing linear stuff. For finite dimensional spaces, when we fix basis on ...
2
votes
1answer
21 views

Rewriting matrix transformation as standard matrix

I'm following a textbook chapter on matrix transformations, and one of the examples seems off. Would this not actually be: $$T\begin{pmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix}\end{pmatrix} = ...
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0answers
7 views

Behavior of norm on matrix algebra under multiplying matrix of scalars

If $A$ is a Banach algebra and I equip $M_n(A)$ with the norm $\|[a_{ij}]\| = \max_i\sum_j \|a_{ij}\|$, do I have $\|Z[a_{ij}]\|\leq\|Z\| \|[a_{ij}]\|$ when $Z$ is a matrix with complex scalar entries ...
0
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0answers
16 views

optimizing over a set of symmetric matrices

I need to minimize a complicated loss function, $f\left(\Lambda\right)$ over a set of symmetric matrices, $S_{p}$ of dimension p, such that all the eigenvalues of $\Lambda \in \left[0,1\right]$. I ...
0
votes
1answer
21 views

How do I solve for A in the matrix equation $A - B(A./C) = D$?

I've got $A - B(A./C) = D$, and I want to solve for $A$.* $A$ is an unknown 2x1 vector, $B$ is a 2x2 matrix, $C$ is a 2x1 vector, and $D$ is a 2x1 vector. *The notation $A./C$ means each element of ...
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0answers
25 views

Normalization of matrix with constraints

I have a matrix of the form: $$ \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right) $$ The rows of the matrix add up to 1. That is $ a+b+c = 1$, ...
1
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1answer
12 views

Characteristic Polynomial of transition matrix of $n$-cycle

Let $P$ be the transition matrix of the deterministic random walk on the cycle $C_n$, i.e. $P \in \{0,1\}^{n \times n}$ with $$P_{i,j}=1\quad \text{ iff }\quad j=i+1 \mod n.$$ My guess is that ...
2
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0answers
57 views

The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
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1answer
19 views

Group action of $GL(2, F)$ on the projective line $P(F)$

I refer to section 8.3, page 119 of Algebra, A Computational Introduction by John Scherk. It is about group action of $GL(2, F)$ on the projective line $P(F) = F \cup \{\infty\}$. Given a matrix ...
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0answers
10 views

Change from one cartesian co-ordinate system to another by translation and rotation.

There are two reasons for me to ask this question: I want to know if my understanding on this issue is correct. To clarify a doubt I have. I want to change the co-ordinate system of a set of ...
0
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0answers
32 views

Skew-symmetric matrix property

This page gives the relation $\left[R\vec{\omega}\right]_{\times}=R\left[\vec{\omega}\right]_{\times}R^{T}$ where $R$ is a DCM (Direction Cosine Matrix), $\vec{v}$ is the angular velocity vector and ...
0
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2answers
25 views

Question on how a matrix is calculated from an example [on hold]

I have the following laplacian matrix given to me in a textbook. In the textbook, the matrix calculation is always done from the 3 x 3 matrix (the methods I learnt makes me cut the matrix further ...
0
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1answer
27 views

Matrix Multiplication, Trace and Integration

Let $\omega(x)$ be a $p\times 1$ vector-valued function defined on a random variable $X$ with CDF $F$. Now define $$V:=\int \omega(x)[\omega(x)]^T dF(x).$$ Then define $\gamma$ as follows. $$ \gamma ...
2
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0answers
25 views

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2n−1$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$.

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2n−1$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$. How to prove this result? (I found this statement while reading ...
0
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0answers
17 views

Product of a Householder transformation and reflection through the origin in 3 dimensions

This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$ $v^T\cdot w=0$, and the Householder transformation ...
0
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1answer
30 views

Prove that multiplying an elementary matrix to a matrix can produce the same effect as an elementary row operation.

Elementary row operations: 1) Interchange any two rows of the matrix 2) Multiply every entry of some row of the matrix by the same nonzero scalar 3) Add a multiple of one row of the matrix to ...
2
votes
1answer
14 views

Find the matrix representation in the standard basis for either rotation by an angle $\theta$ in the plane perpendicular to the subspace

Find the matrix representation in the standard basis for either rotation by an angle $\theta$ in the plane perpendicular to the subspace spanned by vectors $(1,~1,~1,~1)~and~(1,~1,~1,0)$ in ...
1
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1answer
26 views

Subspace of symmetric commuting matrices

I am given $W$ a subspace of real $n$-dimensional matrices which are symmetric and pairwise commuting. I have to prove that $dim(W) \leq n$. I have read some facts about commuting matrices over an ...
8
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4answers
981 views

Matrix inverses - Why are they derived the way they are?

Note that this is not a question of how, but why. I know the mechanics of it, but this is the first thing i've come across that truly seems like magic, rather than a rigorous mathematical process. ...
1
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1answer
18 views

How to determine fewest matrix rows that have entries for all columns?

My apologies for the poor title and description, it's been a long time since I had linear algebra (or any formal math class). Given the following example matrix: \begin{matrix} & W & X & ...
1
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1answer
23 views

Proving that a process is not a Markov chain by using definition.

I want to prove that the queue length at a store is not a Discrete Parameter Markov Chain (DPMC). Now I have the equation: $$Q_k = (Q_{k-1} - 1) + V_k$$ $Q_k$ is the queue length at time instant ...
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2answers
30 views

What am I doing wrong? - Change of basis matrix

Problem: Let $\alpha$ be the standard basis of $\mathbb{R}^3$ and let $\beta = \left\{(1,0,0), (1,1,0), (1,1,1)\right\}$ be another basis. Consider the linear map $T: \mathbb{R}^3 \rightarrow ...
0
votes
1answer
33 views

proof of the singular-values of orthogonal matrix

What is a simple and intuitive proof that the singular-values of orthogonal matrix $A$ is $1$?
0
votes
1answer
48 views

Condition number of $A^TA$

if $n \times n$ full rank matrix $A$ has condition number $\kappa$, what would be the condition number of $A^TA$? Preferably If the derivation includes the following definition of $\kappa$: $$ \kappa ...
1
vote
3answers
48 views

Maximum determinant of $3 \times 3$ matrix

Good one guys! I'm studying to the maths olympiads in my college and I ran to the following problem: What is the possible matrix $3 \times 3$, that you can write using digits from $0 $ to $9$, (you ...
0
votes
1answer
10 views

Isometry property of semi-orthogonal matrices

I've got a question concerning semi-orthogonal matrices. In their book 'Matrix Algebra', Abadir and Magnus define a semi-orthogonal matrix as a matrix A satisfying one of the two equations: $A^T\cdot ...
0
votes
1answer
18 views

Solving Lyapunov equation for unknown A matrix and known P matrix

I need to solve the Lyapunov equation $A'P+PA+Q+PBR^{-1}B'P=0$ for matrix A. Note that usually the equation is solved to get unknown P matrix. But instead of the usual problem which can be solved ...
0
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2answers
64 views

2014 IMC first problem first day (eigenvalues of a product of symmetric matrices).

This was the first problem of the IMC 2014. Let $A$ and $B$ be two $n\times n$ symmetric matrices with real entries which have all their eigenvalues strictly larger than $1$. Prove all the ...
0
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0answers
11 views

Eigenvalues of normalized adjacency matrix

Can anyone introduce some references on the eigenvalue estimation of normalized adjacency matrix, i.e., $W=D^{-1}A$ ($D$ is the degree matrix and $A$ is the adjacency matrix of the corresponding ...
3
votes
1answer
78 views

$A^tA-AA^t$ in Mathematical Physics

In very different contexts of mathematical physics (rigid body mechanics, fluidodynamics, general relativity, quantum field theory,...) I have come across the following expression: $$ A^tA-AA^t, $$ ...
0
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0answers
27 views

Efficient Test For Commuting Matrices

I know that if $A$ and $B$ are two Hermitian matrices, then $A B= B A$ if and only if their eigenspaces coincide [1]. In order to apply this test one need to compute eigenvectors of both $A$ and $B$ ...
3
votes
0answers
22 views

Quadratic form and matrix

We know quadratic form $f(x_1,x_2)= a_{11} x_1^2 + 2 a_{12} x_1 x_2 + a_{22} x_2^2$ is non-negative for all $x_1,x_2 \in \mathbb{R}$ iff matrix $(a_{ij})_{2 \times 2}$ is semi-positive defined. My ...
1
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1answer
12 views

Bound on maximum of product of matrix and vector

I need to bound the absolute maximum of each entry of a matrix-vector product: $\max_{|x|_{1}=1} |Ax|_{\infty}$ I tried to pose this in terms of the induced infinity norm of $A$, as in ...
0
votes
0answers
12 views

About a property of the upper triangular projection of a matrix

I need a hand checking that a property about the upper triangle projection of an infinite matrix holds. $\bullet$ Let A be an infinite matrix $A=(a_{ij})_{i\geq 1\;j\geq 1}$. We define its upper ...
5
votes
3answers
286 views

How to find the limit of this matrix function

Let $A$ be $n\times n$ real symmetric matrix that is positive definite. Let $x\in\mathbb{R^n}, \space x\ne 0$. Prove that the following limit $$ \lim_{m\to\infty}\dfrac{x^TA^{m+1}x}{x^TA^{m}x} $$ ...
1
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3answers
168 views

Symmetric matrices and eigenvalues

If the eigenvalues of a symmetric matrix $A$ are greater than 0, show that $v^{\top}Av > 0$ for every $v \ne 0$ I am trying to prove this as follows: If $v$ is an eigenvector of $A$, then $Av ...
0
votes
1answer
20 views

What can I assume, when given a matrix with information about its eigenvalues but not its action?

Basically, I've had to use linearity a couple of times yesterday and today, in order to write up a few proofs. But I notice that I am only given information such as positivity conditions and ...
5
votes
3answers
103 views

If det $A = 0$ and $\det B \neq 0$ then show that $abc = -1$

This has been hurting my head for a while now.... If $$ \det\begin{bmatrix}a&a^2&1+a^3\\b&b^2&1+b^3\\c&c^2&1+c^3\end{bmatrix}=0 $$ And $$ ...
1
vote
0answers
38 views

Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
0
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0answers
26 views

A subgroup of special linear group

Does anybody know if the subgroup of diagonal and antidiagonal matrices of $SL(n,F)$ has been given a particular name? By $SL(n,F)$ I mean $n \times n$ matrices over a field $F$ with determinant 1. ...
1
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1answer
27 views

Find the maximum value of this form

Let $A,B$ be $n\times n$ real symmetric matrices such that $B$ is positive definite. Show that $G$ defined below attains a maximum value at an eigenvector related to $A$ and $B$. Also find the ...
0
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0answers
15 views

Multivariable polynomial matrix representations

This is a follow-up to matrix representation of parabola and matrix representation to generate monomials. I found a method to build such matrices to implement this type of functionality for one ...
0
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0answers
19 views

What is pseudospectra of matrix polynomials? .

What is pseudo spectra of matrix polynomials? Please guide me with some example or some reference regarding it. Thank You!
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3answers
39 views

Evaluating a function at a point where $x =$ matrix.

Given $A=\left( \begin{array} {lcr} 1 & -1\\ 2 & 3 \end{array} \right)$ and $f(x)=x^2-3x+3$ calculate $f(A)$. I tried to consider the constant $3$ as $3$ times the identity matrix ($3I$) but ...
-2
votes
2answers
46 views

For what values of $a$, $b$, and $c$ the above system has: One solution. Infinitely many solutions. No solutions.

I am stuck with this now, I tried reducing the matrix to row echelon form, but it gets a bit hard. Is there not a simpler way? The system is: \begin{align*} a x + b y − 3 z &= −3\\ −2 x − b y + ...
3
votes
1answer
50 views

Scaling a svg image while keeping the offset position.

I have an svg image of a map that i have to scale up to make it zoom in. Javascript has a function to scale up SVG images. However the svg scale function uses the upper left corner as center when ...
0
votes
3answers
35 views

Matrix Multiplication: Both ways okay?

Say I have two matrices $A$ and $B$ where $A$ has dimensions of $1 \times 2$ ($1$ row, $2$ columns) and $B$ has dimensions of $2 \times 3$ ($2$ rows, $3$ columns) When you multiply these like so $(A ...
0
votes
0answers
21 views

PCA of the large symmetric almost-diagonal matrix

I was doing factor reduction of the correlation matrix of the special form $\rho_{ij}=\rho+(1-\rho)e^{-\beta |i-j| }$, with $i,j \le n=100$, $\rho \ll 1$ and $\beta \le 1 $. $$ \begin{bmatrix} ...