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), ...

learn more… | top users | synonyms (2)

0
votes
1answer
34 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
votes
0answers
6 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
61 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
votes
0answers
23 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
18 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
votes
0answers
5 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
268 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
157 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
92 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
votes
0answers
20 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
votes
0answers
21 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
vote
1answer
25 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
votes
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
votes
0answers
17 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!
1
vote
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
43 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 + ...
2
votes
1answer
42 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
votes
0answers
13 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
50 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
23 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
votes
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
174 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
15 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
41 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 ...
0
votes
1answer
33 views

Question on normal matrices

Hello all I was given this question in my linear algebra class which I have tried to solve but to no avail, and I would really appreciate any help. I am given a matrix $ A \in M_{nxn}(C) $ and am ...
1
vote
4answers
43 views

How to find a real matrix with complex eigenvalues,

Give a $2 \times 2$ real matrix $A$ with eigenvalues $2+3i$, $2-3i$. I would like hints only. So far, I've been trying get somewhere with $\det[A-(2+3i)I] = 0$ and $\det[A-(2-3i)I] = 0$; which ...
1
vote
1answer
22 views

Proving Properties of Discrete Time Markov Chain mathematically

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
vote
0answers
28 views

Is it okay to perform the same row operation twice on opposite rows?

I am trying to find the inverse of the following matrix: 1 2 3 2 1 4 1 0 2 I draw the identity matrix next to it and start performing row operations. ...
1
vote
3answers
78 views

Eigenvalues of matrix $A^TA+I$ are real and greater than 1?

In this paper, the author states that the eigenvalues of the matrix $A^TA + I$ are real and greater than 1, since $A^TA$ is symmetric positive definite. But why?
0
votes
1answer
20 views

Bounding the smallest eigenvalue of symmetric matrix product

Let $X = ABA^T$ where $B \in \mathbb{R}^{p \times p}$ and $B$ is positive definite matrix and $A \in \mathbb{R}^{q \times p}$ so that $X \in \mathbb{R}^{q \times q}$. My question is concerning an ...
1
vote
2answers
28 views

Get the camera transformation matrix (Camera pose, not view matrix)

Let's say that I have an object and a camera (its representation) in a 3D world coordinate system. I have the camera pose to see the object (rotation matrix and translation (eye position)). If I apply ...
0
votes
0answers
15 views

What formula would I use for a four factor prioritization method where the factors are summed and ranked?

We are developing a way to prioritize system issues. Our current ranking is 1 - 5, but that becomes rather flat when dealing with a couple hundred issues. In our new method, we have four factors in ...
3
votes
0answers
43 views

When does a matrix have short vectors in its kernel?

Consider an $n$ by $n$ matrix $M$ whose elements are in $\{0,1\}$, say. Now consider all vectors $v \in \mathbb{Z}^n$. Is there any mathematical property of $M$ which expresses when the kernel of ...
1
vote
1answer
70 views

is this an orthogonal matrix?

$T$ is a $4\times 4$ real matrix, and obeys $$T^\dagger \left( \begin{array}{cccc} a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a^2 & 0 \\ 0 & 0 & 0 ...
0
votes
0answers
36 views

Multiple Matrices Multiplications [on hold]

I am trying to write an algorithm based on a paper I came across (DOI: 10.1109/TNN.2009.2039226). I need to multiply 5 matrices: AxBxCxDxE. A is L1xL1, B is L1xL2, C is L2xL2, D is L2xL2, E is L0xL0 ...
0
votes
0answers
12 views

Low rank approximation of a set of matrices with equal residuals

Given a set of $n \times n$ matrices $\boldsymbol{H}_i$ ($i=1,2,\cdots,N$, $N \geq 2$) and an interger $k$ ($k<n$), is there a non-iterative way to solve the following problem if the solution ...
1
vote
0answers
7 views

Finding minimal projections in subalgebra generated by a given set

Consider the set of complex matrices $\mathbb{C}^{n\times n}$ for some set. Suppose we have a set $\{A_1,\ldots, A_n\}$ of Hermitian matrices. We want to find minimal projections in the subalgebra ...
0
votes
0answers
21 views

Specific vectors in the null space of a complex matrix

I'm dealing here with the following problem: I have a complex matrix $F$, having size $N \times M$, with $N<<M$. I can easily compute a basis of the null space of $F$, i.e. the space of vectors ...