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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2 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.
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2answers
33 views

How many techniques are there to compute the determinant of any NxN matrix?

How many techniques are there in total to compute the determinant of any NxN matrix?
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2answers
35 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 ...
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1answer
25 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 ...
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4answers
35 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 ...
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0answers
12 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 ...
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0answers
26 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
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3answers
75 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?
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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 ...
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2answers
24 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
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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
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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
69 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 ...
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0answers
34 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 ...
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0answers
10 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 ...
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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 ...
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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 ...
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4answers
503 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
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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
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2answers
31 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 ...
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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 ...
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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 ...
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2answers
29 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 ...
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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
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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
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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
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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
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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
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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 ...
3
votes
1answer
28 views

Find the inverse of a specific Vandermonde matrix

Let $$ V=\begin{bmatrix} 1& 1& 1& \cdots& 1 \\ 1& \xi& \xi^{2}& \cdots& \xi^{n-1} \\ 1& ...
0
votes
1answer
20 views

The Maximum Eigenvalue of $F\mathrm{max(B)}F^T - FBF^T$

$F$ is a $b \times n$ real matrix. $B$ is a $n \times n$ real matrix, constructed by $B = w^T w$, where $w$ is a row vector with strictly positive real numbers, and clearly $B$ is a rank 1 matrix. ...
1
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1answer
34 views

Does $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({AY+YA}^T) = \{0\} $ imply $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({CAY}) = \{0\} $?

Given $ \mathbf{Y}=\mathbf{Y}^T \in \mathbb{R}^{n\times n} >0, \mathbf{A} \in \mathbb{R}^{n\times n} $ Hurwitz, $ \mathbf{C} \in \mathbb{R}^{m\times n}, \mathrm{rank}(\mathbf{C})=m,\ m \le n $, I ...
13
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2answers
405 views

Are eigenvalues of the limit of a sequence of matrices limits of eigenvalue sequences?

Let $\{A_n\}\in \mathbb{R}^{m\times m}$ be a sequence of symmetric matrices such that $A_n\to A$ as $n\to \infty$, i.e. $\lim_{n\to \infty}a_{ij}(n)=a_{ij}\ \forall 1\le i,j\le m$ where ...
0
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0answers
20 views

Derivation of energy function

Given the following energy function $E(d)$ (also found here on page 3): $$ E_d = \sum_{x,y \in \Omega} \left(d_{x,y} - \hat{d}_{x,y}\right)^2 + \lambda \sum_{x,y} \left( ...
3
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1answer
37 views

How to change of basis from 3 points

I'm a computer scientist and I'm not very good with mathematical stuff... I have got 3 points A, B, C that doesn't create an orthogonal space. I have the coordinates of those 3 points in two 3D ...
1
vote
1answer
47 views

Construction of a matrix over $ \{-1,0,1\} $

Let $ Z=(z_{ij}) $ be a $ (n,n)$-matrix, for which: $ z_{ij} \in \mathbb{R}; $ $ z_{ij}= -z_{ji} $ for $ i,j=1, \dots , n; $ $ \sum_{j=1}^n z_{ij} = 0 $ and $ \sum_{j=1}^n |z_{ij} |>0 $ for $ ...
2
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1answer
34 views

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row?

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row(uniform distribution)? What sort of algorithm should I use to do this task? Brute Force algorithm- ...
0
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0answers
26 views

Is the converse of the Spectral Theorem true?

In the book by Friedberg, Insel and Spence, symmetric matrices are orthogonally diagonalizable, and over the complex number field, normal matrices are orthogonally diagonalizable -- this is all from ...
0
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1answer
14 views

For which value of m are these 3 vectors linearly dependent?

In one of my revision worksheets there is a question which goes as follows: The vectors u=mi+j+k, v=i+mj+k and w=i+j+mk, where m is a real constant, are linearly dependent for either m=0, m=1, m=2, ...
0
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1answer
32 views

Transition Probability matrix for on/off swtiches

Lets say we have $5$ on/off switches. On state is designated by 1 and off state is designated by $0$. Then, $\{ W_k^i, k =0,1,2,\ldots\}$ is a discrete parameter Markov chains (DPMCs) for each ...
0
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5answers
29 views

Find a matrix which maximizes expression

Assume I have column vectors $x,y\in\mathbb{R}^n$, and the following expression $$ A\in M_n(\mathbb{R}),\ |\det A|\leq 1,\ K(A) = x^tAy $$ How can I find the matrix $A$ that maximizes expression ...
0
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0answers
18 views

Compact representation of the following matrix

I have a matrix that has the following structure \begin{bmatrix} a_1(1) & 0 & a_2(1) & 0 \\ 0 & a_2(1) & 0 & a_1(1) \\ a_1(2) & 0 & a_2(2) & 0 \\ 0 & a_2(2) ...
3
votes
1answer
51 views

A Real Matrix, its Kernel and Image

This is an old exam problem: For an $m \times n$ real matrix $A$, define $\ker A = \{x \in \mathbb{R}^n \mid Ax=0 \}$ and $\operatorname{Im} A = \{Ax \mid x \in \mathbb{R}^n \}$. Show that for all $b ...
2
votes
2answers
33 views

Does the lowest diagonal element of a real symmetric matrix form an upper bound to the lowest eigenvalue?

If I have a real symmetric matrix, is it possible to look at the lowest diagonal element and then claim that the lowest eigenvalue of the matrix must be less than or equal to that diagonal element? I ...
0
votes
4answers
98 views

I have discovered a way to calculate the absolute value (area,volume, etc) of a n-dimentional shape, using it's coordinates only, how do I publish it?

Firstly, I want to preface by saying that I am no experience with the maths community at all, however I did take Maths and Further Maths for my A-Levels. What I have discovered is a way of using ...
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2answers
32 views

How to determine if matrices are similar?

Trying to teach myself some Linear Algebra, now trying to study about similar matrices concept, but i am having some trouble (maybe because i am trying to teach myself), found a question online and i ...
0
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0answers
15 views

Algorithm for getting Markov chain given the complex eigenvalues

Given real and complex eigenvalues (occurring in conjugate pairs) how to get a single instance of a Markov Chain which has these eigenvalues. I know the Markov chain is not unique as eigenvectors are ...
0
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0answers
33 views

a matrix metric

Let $U_1,...,U_n$ and $V_1,...,V_n$ be two sets of $n$ matrices of the same size. We'll denote $E(U_i,V_i)= \max_v \, |(U_i -V_i)v|$ (max over all the quantum states), $U=\prod_i U_i$ and $V=\prod_i ...
0
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2answers
27 views

Algorithm for real matrix given the complex eigenvalues

Given complex eigenvalues (occurring in conjugate pairs) how to get a single instance of a real matrix which has these eigenvalues. I know the matrix is not unique as eigenvectors are not fixed but in ...