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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1
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1answer
19 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
14 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
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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 ...
0
votes
2answers
42 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
31 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
34 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
13 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
6 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 ...
1
vote
1answer
27 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
42 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
26 views

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

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
171 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
18 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
14 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
6 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
54 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
31 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
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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
19 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
27 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
14 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
35 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
11 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 ...
0
votes
0answers
4 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 ...
11
votes
4answers
508 views

Solving a system of non-linear equations

Let $$(\star)\begin{cases} \begin{vmatrix} x&y\\ z&x\\ \end{vmatrix}=1, \\ \begin{vmatrix} y&z\\ x&y\\ \end{vmatrix}=2, \\ \begin{vmatrix} z&x\\ y&z\\ ...
1
vote
2answers
27 views

Reduced row echelon form without introducing fractions at any intermediate stage

How can I reduce this matrix to reduced row echelon form but without using fractions in intermediary steps (I can use them in elementary row operations just not in the results in the matrix) $$ ...
1
vote
2answers
32 views

Proving existence and uniqueness of a matrix,

Let A be nxn with real coefficients and assume that it has n distinct eigenvalues, and all eigenvalues are positive real numbers. Let k $\ge$3 be an odd integer. a) Prove there exists a unique real ...
0
votes
0answers
9 views

Calculate camera view and projection matricies from projected points

I’m stuck on a project for a client.. I need to find the answer to this to proceed: Given (n) coordinates in 3D space and (n) corresponding coordinates in 2D space as projected onto a camera’s image ...
0
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0answers
34 views

Is there a function that defines the trace of a matrix A divided by its determinant?

Given a matrix $A$ that is $n \times n$ with a non-zero determinant does there exist a function $f(n,A)$ in any field of mathematics such that: $$ f(n,A) = ...
2
votes
2answers
42 views

What is the correct way to write this matrix equation?

Given an $n \times m$ matrix $X$ and $m \times m$ matrix $A$, I would like to define the vector $y$ as $$y_i = X_{i,*} A (X_{i,*})^T$$ where $X_{i,*}$ is the $i$th row of $X$. Is there a simpler ...
0
votes
2answers
30 views

Block Matrix Zero Determinant Implication?

Recently I've been working with a number of square (order of 2n) matrices whose determinants are zero. That is, $$\det\begin{bmatrix}A&B\\C &D \end{bmatrix} = 0$$ where each of A,B,C, and D ...
0
votes
1answer
8 views

Point within a Cube in 3D environment

I have a cube in 3D space with 8 corner points with their X,Y,Z Coordinates. I know how to test if any given point lies inside a cube by Comparing their coordinates to be greater or smaller than the ...
1
vote
1answer
12 views

How does this form of Poincare's inequality for self-adjoint matrices hold?

I'm reading "Introduction to Matrix Analysis and Applications" by Hiai and Petz, and they state Theorem 1.26 ("Poincare's Inequality") as follows: Let $A\in B(H)$ be a self-adjoint operator with ...
1
vote
1answer
10 views

Hilbert-Schmidt matrix and square sumable

I am reading a note but I encountered two concepts that I don't know their definitions. 1- Hilbert-Schmidt matrix I searched but I only saw measures and such things, I want definition of ...
0
votes
0answers
8 views

AAK theorem for finite dimensional Hankel matrix

Does the AAK theorem hold for finite dimensional Hankel matrix? Or maybe similar analysis exists? (From a quick look of the proof, it seems like the AAK solution has to be infinite dimensional ...
0
votes
0answers
23 views

Given a matrix M, is there a name for the matrix MM^T?

One can make a symmetric square matrix out of any m-by-n matrix $M$ by computing the matrix $MM^T$ (or $M^T M$). Is there a name for this operation? I want to call it "symmetrizing" the matrix, but I ...
1
vote
0answers
22 views

transofrmations (a,b,c) to (x,y,z)

I'm not 100% sure linear algebra will crunch this problem, but hopefully so. This may just be a case of matrices, which would be good cause I like those. Imagine we have a robot with a camera ...