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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8 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
28 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 ...
0
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0answers
10 views

Move 2 vectors (a line) in line with an arbitrary third vector

Related question on Blender SE: http://blender.stackexchange.com/questions/33162/move-an-edge-in-line-with-an-empty I have been trying a couple of things but I am somewhat stuck. I am no math genius ...
2
votes
1answer
13 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
vote
1answer
24 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 ...
3
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3answers
394 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
16 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 & ...
0
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0answers
18 views

Prove that a process is not a Markov chain 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 ...
1
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2answers
29 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
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1answer
27 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 $1$?
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1answer
36 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
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3answers
43 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 ...
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1answer
6 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 ...
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0answers
14 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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1answer
38 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 ...
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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
73 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: $$ A^tA-AA^t, $$ ...
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0answers
25 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
vote
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
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0answers
6 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
282 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
vote
3answers
162 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
98 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 $$ ...
0
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0answers
32 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
24 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
26 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
18 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 ...
-1
votes
2answers
44 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
47 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
19 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} ...
0
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0answers
14 views

Calculating Cosine Similarity with Matrix Decomposition (matrix multiplication with normalized columns)

To calculate the column cosine similarity of $\mathbf{R} \in \mathbb{R}^{m \times n}$, $\mathbf{R}$ is normalized by Norm2 of their columns, then the cosine similarity is calculated as ...
2
votes
1answer
36 views

Determine the isomorphism class of $\mathbb Z^3 / M$ for the subgroup $M$ of $\mathbb Z^3$generated by $(13,9,2),(29,21,5),(2,2,2)$

The problem seems not so hard. My confusion rise from the statement in the solution above that "This question is equivalent to reducing the matrix via row and column operations". Please see the ...
1
vote
1answer
54 views

Consequences of the positivity condition $v^t A v > 0$ for the eigenvalues of $A$

Let $A$ be an $n \times n$ symmetric real matrix with n distinct eigenvalues $\lambda_1 , ... , \lambda_n$. a) Suppose $v^t(Av)$>0 for all v in $R^n$, v$\ne$0. Show that all $\lambda_i$ are positive ...
1
vote
1answer
24 views

Transition matrix question,

In diagonalizing a matrix A, we use a matrix S, which consists of eigenvectors of A. To compute S, we simply take each eigenvector and write it as a linear combination of the standard basis. So if ...
0
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0answers
28 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz [duplicate]

I have asked this question on mathoverflow also. (my question, I wasn't sure if its ok ask at another similar forum, on stack exchange, but I hope it would reach more people). It is well known how to ...
10
votes
3answers
177 views

Find this Determinant

I have to find this determinant, call it $D$ \begin{vmatrix} \frac12 & \frac1{3}& \frac1{4} & \dots & \frac1{n+1} \\ \frac1{3} & \frac14 & \frac15 & \dots & ...
-4
votes
1answer
19 views

double summations

Assume that w1=0.4; w2=0.5; w3 =0.1. Basing on the following matrix that provides values for xij : 7 4 9 6 4 12 3 2 17: Calculate the following value: 3 3 ∑∑wiwjXij i=1 ...
0
votes
2answers
23 views

Deriving a Formula for the determinant of a block matrix.

This is a follow up question to this. I want to solve the following problem: Let $n \in \Bbb N \space \text{/{0}} \space \text{and} \space n_1,n_2 \in \Bbb N \space \text{such that} \space ...
0
votes
1answer
17 views

Finding change of basis matrix when given two bases as a set of matrices

Find the change of basis matrix between the following bases: $\alpha = \left\{ \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}, \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}, ...
0
votes
3answers
42 views

Book for Linear Algebra and Matrix

my major is Electrical Engineering and I am new in linear algebra and I need to be familiar with matrix theory deeply because of my research topic which is Image Processing. But, I do not know from ...
1
vote
1answer
43 views

When is the matrix $\mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T}$ a symmetric matrix?

let $\mathbf{A}\in\mathbb{R}^{m\times n}$ and $\mathbf{x}\in\mathbb{R}^{n\times 1}$. \begin{equation} \mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T} \end{equation} Can we say that ...
0
votes
1answer
36 views

Matrices inside matrix. Showing $det(M)=det(C)$

Let $n \in \Bbb N \space \text{/{0}} \space \text{and} \space n_1,n_2 \in \Bbb N \space \text{such that} \space n_1+n_2=n$ $$M=\begin{pmatrix}E_{n_1}&B\\O&C\end{pmatrix}$$ where $E_{n_1} ...
0
votes
0answers
7 views

What is pseudospectra of matrix polynomials?

What is pseudospectra of matrix polynomials? Please guide me with some example or some refrence regarding it. Thanks.
-4
votes
2answers
55 views

Crout matrix decomposition [on hold]

In naive terms and step by step, how to to find the determinant of any NxN matrix by using LU Decomposition of Crout's method. Also, discuss its efficiency as compared to other LU decomposition ...
3
votes
3answers
180 views

matrix representations and polynomials

I just investigated the following matrix and some of its lower powers: $$M = \left[\begin{array}{cccc} 1&0&0&0\\ 1&1&0&0\\ 1&1&1&0\\ 1&1&1&1 ...