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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3
votes
1answer
61 views

Powers of $2 \times 2$ matrices expressed in linear form

I recently reopened an old high school math textbook and came upon the matrices unit. Some of the questions were those rewrite-in-linear-form problems: given, say, $M^2 = 2M - I$, express in linear ...
0
votes
1answer
110 views

Diagonalizing a $2\times 2$ matrix

I am told that $A\in M_{2}(\mathbb{R})$ s.t $1-i$ is an eigenvalue of $A$ with corresponding eigenvector $\begin{pmatrix}3+2i\\ 1+3i \end{pmatrix}$. I wish to find a matrix $P$ s.t ...
1
vote
1answer
106 views

determinant of matrix of transformation from Cartesian to orthogonal curvilinear

Let $(x_1, x_2)$ and $(y_1, y_1)$ be two orthogonal coordinate system with unit vectos $(\hat i_1, \hat i_2)$ and $(\hat e_1, \hat e_2)$ respectively defined by the $x_1 = x_1(y_1,y_2)$ and $x_2 = ...
9
votes
1answer
1k views

Does equality of characteristic polynomials guarantee equivalence of matrices?

I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and ...
1
vote
2answers
40 views

An example of a $2\times2$ matrix $A$ without zero entries and with eigenvalues $\lambda_{1}=3,\lambda_{2}=-4$

I am trying to do an exercise that asks : Find an example of a $2\times2$ matrix $A$ without zero entries and with eigenvalues $\lambda_{1}=3,\lambda_{2}=-4$ I am having trouble thinking of a ...
1
vote
0answers
92 views

how to Evaluate integral of density of Wishart matrix

Let $X_1 \cdots X_N$ are $N$ number of $m$ Dimensional Independent Complex Gaussian Random vectors Such that: $$ X_j \sim \mathcal{N}(\mu,\Sigma)\; \forall \;j=1 \cdots N$$ Let ...
3
votes
1answer
89 views

Characteristic equation for 2-nd order ODE

Given a differential equation $\dot x = Ax$, $x \in \mathbb{R}^n$ we define its characteristic equation as $\chi(\lambda) = \det (\lambda I - A)$. Consider now the second order ODE $$ \ddot x + A x ...
1
vote
1answer
116 views

V is a vector space such that $V = A\oplus A^\perp$ also $V = A \oplus C$ then can we say that $A^\perp = C$?

I have a vector space $V$ such that $V = A\oplus A^\perp$ i.e. $V$ is a direct sum of its subspace $A$ and orthogonal complement of $A$. Suppose we also have $V = A \oplus C$ Then can we say that ...
3
votes
3answers
1k views

What is the importance of determinants in linear algebra?

In some literature on linear algebra determinants play a critical role and are emphasized in the earlier chapters. (See books by Anton & Rorres, and Lay). However in other literature it is totally ...
0
votes
1answer
53 views

Modify a matrix so that it contains continuous intervals

I'm using MatLab. I am new to it, and new to matrix calculation, too. Consider the following matrix : ...
7
votes
4answers
9k views

Similar matrices have the same eigenvalues with the same geometric multiplicity

Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities. Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices ...
0
votes
1answer
126 views

Finding the mean (or pattern) of a set of related values in a matrix discarding those that are too different.

this is my first question here so I am not sure if it is a valid one. I am currently facing a mathematical problem that is not so hard to solve, but given the accuracy i need in the results I would ...
0
votes
2answers
385 views

convert values from one coordinate system (x,y) to another coordinate system (x', y')

Following is a graph that contains both coordinate systems (x,y) and (x',y'). x, y, x', and y' are all axes ...
5
votes
3answers
567 views

Endomorphisms, their matrix representations, and relation to similarity

This question is really several short general questions to clear up some confusions. We'll start with where I'm at: An endomorphism $\phi$ is a map from a vector space $V$ to itself. After choosing a ...
-2
votes
3answers
3k views

Condition for commuting matrices

Let $A,B$ be $n \times n$ matrices over the complex numbers. If $B=p(A)$ where $p(x) \in \mathbb{C}[x]$ then certainly $A,B$ commute. Under which conditions the converse is true? Thanks :-)
1
vote
2answers
93 views

Is the diagonalization of A Invertable?

There is a theorem which says that given a diagonalizable matrix $A$ such that $P^{-1}AP=D$ if $D$ is invertible then A is invertible. I suspect that the other direction isn't true, but I can't think ...
21
votes
11answers
8k views

What is the usefulness of matrices?

I have matrices for my syllabus but I don't know where do they find their use. I even asked my teacher but she also has no answer. Can anyone please tell me where are they used? And please also give ...
0
votes
1answer
37 views

Which matrix operation should I use.

The title is quite vague, but I don't see how to phrase it. I'm new to MatLab and have very little experience with matrix calculation. Suppose a matrix "a" : ...
2
votes
2answers
287 views

Why is the matrix representing a non-degenerate sesquilinear form invertible?

Let's consider a finite-dimensional vector space $E$ on the field $\mathbb{K}$ (where $\mathbb{K}=\mathbb{C} \ \text{or}\ \mathbb{R}$) and a sesquilinear (or bilinear if $\mathbb{K}=\mathbb{R}$) form ...
0
votes
1answer
237 views

How do I handle image gradient calculation at the edge of images?

The image gradient is the rate of change over any given pixel of an image, either in the horizontal or vertical direction. An image can be thought of as a large matrix of values [0, 255]. A common ...
1
vote
2answers
266 views

Polynomial vector space

What does the following mean -- "The Jordan Canonical Form of the operator $w{d\over dw}$ acting on the complex vector space of polynomials in $w$ of degree less than $n$"? Thank you.
3
votes
0answers
182 views

Does matrix convergence in $L^p$ imply convergence of the eigenvalues in $L^p$?

Let $A_n(x)$ be a sequence of symmetric matrix functions that converges in $L^p(\Omega)$ to $A(x)$. Is it true that the eigenvalues of $A_n(x)$, or a subsequence of these, converge to the eigenvalues ...
3
votes
5answers
118 views

calculate generally the determinant of $A = a_{ij} = \begin{cases}a & i \neq j \\ 1 & i=j \end{cases}$

calculate generally the determinant of $A = a_{ij} = \begin{cases}a & i \neq j \\ 1 & i=j \end{cases} = \begin{pmatrix} 1 & a & a & · & a \\ · & · & · & · \\ a ...
0
votes
6answers
322 views

positive definite quadratic form

Is $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ positive definite? Approach: The matrix of this quadratic form can be derived to be the following $$M := \begin{pmatrix} 1 & ...
0
votes
1answer
80 views

How to find every possible scalar product on $V$

I am given the following task: "For which $a$, $b \in \mathbb{R}$ there exists a scalar product, such that $$A = \left( \begin{matrix} 0 & 0 & a b \\ 1 & 0 & a \\ 0 & 1 & ...
0
votes
0answers
73 views

Elementary matrix operations on group

Given a matrix whose every element is an element of group Sn (Symmetric group of n objects). I want to apply Gaussian Elimination to convert it into row echelon form. I need to find out linearly ...
4
votes
1answer
95 views

Is there a nice way to classify the ideals of the ring of lower triangular matrices?

Suppose $T$ is the subset of $M_2(\mathbb{Z})$ of lower triangular matrices, those of form $\begin{pmatrix} a & 0 \\ b & c\end{pmatrix}$. So $T$ is a subring. Now I know that the ideals of ...
1
vote
1answer
88 views

Orthogonal fitted values

I have two regression models $$Y=X\beta+\varepsilon,\quad \beta\in\mathbb{R}^k$$ $$Y=Z\alpha+u\quad \alpha\in\mathbb{R}^m$$ it is known that using OLS estimates $\hat{\beta},\hat{\alpha}$ fitted ...
2
votes
4answers
196 views

Creating Unique Values based off Two Sets of Sequential Integers

First off, I apologize if this is the wrong board. I'm a heavy StackOverflow user, and this is technically a programming question (or at least serves programming use), but I find it to be based moreso ...
7
votes
3answers
540 views

Trace of powers of a nilpotent matrix

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be 0, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like to ...
2
votes
1answer
66 views

Is $A^{q+2}=A^2$ in $M_2(\mathbb{Z}/p\mathbb{Z})$?

I'm wondering, why is it that for $q=(p^2-1)(p^2-p)$, that $A^{q+2}=A^2$ for any $A\in M_2(\mathbb{Z}/p\mathbb{Z})$? It's not hard to see that $GL_2(\mathbb{Z}/p\mathbb{Z})$ has order ...
1
vote
1answer
128 views

Intersection of Generalized doubly stochastic matrix set and Orthogonal matrix set

The definition for doubly stochastic matrix can be found here. We say a square matrix $A$ is a Generalized doubly stochastic matrix if the sum of each rows and columns of $A$ all equals 1. But A ...
1
vote
2answers
538 views

Positive Semi-Definite matrices and subtraction

I have been wondering about this for some time, and I haven't been able to answer the question myself. I also haven't been able to find anything about it on the internet. So I will ask the question ...
1
vote
1answer
503 views

a matrix inverse laplace transform problem

Let $\mathcal {L}^{-1}[\cdot]$ be an inverse Laplace transform. Let $A$ be a square matrix, and $I$ an identity matrix. Based on the fact that $\mathcal {L} ^{-1} [{(sI-A)}^{-1}] = e ^{tA}$, how can ...
1
vote
3answers
573 views

Characteristic polynomials of powers and sums of matrices

If I know the characteristic polynomial of a matrix $A$, what can I know about the charpoly of $A^2$? And if I have the charpolys of $A$ and $B$, what can I know about the charpoly of $A+B$? I'm ...
2
votes
0answers
229 views

Cholesky decomposition for sparse matrix

I have a matrix that is composed of small block diagonal matrices. For example: $$ M = \left[ \begin{array}{ccc} \Sigma & \Psi & \Psi \\ \Psi & \Sigma & \Psi \\ \Psi & \Psi ...
1
vote
1answer
316 views

The real part of a matrix under similarity transformation

I have a question regarding the real part of some matrix A, defined as $$ Re\{A\} = \frac{1}{2}\left(A + A^\dagger \right).$$ Where $A^\dagger$ denotes the Hermitian conjugate. One can also assume ...
1
vote
1answer
91 views

How do I determine whether a function is operator monotone on a given interval?

Just to avoid confusion, a function is called matrix monotone in an interval $[a, b]$ if $A - B \geq 0$ implies $f(A) - f(B) \geq 0$ for any Hermitian Matrices $A, B$ (we can restrict to finite ...
1
vote
1answer
682 views

sufficient and necessary conditions for convergence of geometric series of matrices

A is a square matrix with the following properties: 1. the diagonal elements are zero. 2. every element in the same row shares the same positive value. What is the sufficient and necessary ...
2
votes
4answers
86 views

Derivative of $\|Xa\|_2 $ with respect to $X$

Can someone give me the answer to the following expression? $\frac{\partial}{\partial X}\|Xa\|_2 =?$ $X$ is a square matrix and $a$ is a vektor of the apropriate size. $\|\cdot\|_2$ is the euclidean ...
0
votes
1answer
117 views

Is there any closed-form expression to calculate each element of the inverse of a matrix?

Considering a generic square matrix $A=(a_{i,j})$ we want to compute its inverse $A^{-1}=\left[a^{(-1)}_{i,j}\right]$. Is there a way to express each $a^{(-1)}_{i,j}$ using a closed form expression?
8
votes
1answer
391 views

Dimensionality of null space when Trace is Zero

This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216). I've succeeded in proving the first three parts, but the most ...
9
votes
2answers
788 views

Augmented Reality Transformation Matrix Optimization

i am a software developer, i'm working on an Augmented Reality system. I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast. Here's the situation: ...
2
votes
1answer
178 views

rank for the matrix of concatenating all $N \times N$ permutation matrics

Consider all $N\times N$ permutation matrix $\{M_1,M_2,\ldots,M_{N!}\}$ Define $S_N$ as concatenating each $\operatorname{vec}(M_i)$ as $S_N$'s $i$th column Is there any convenient way to calculate ...
2
votes
2answers
273 views

“Splitting” determinant of a matrix

There is a thing that I don't understand about how the determinant of a matrix could be split this way: $$ \begin{vmatrix} a & b\\ c & d \end{vmatrix}= \begin{vmatrix} a & 0\\ c ...
2
votes
5answers
2k views
2
votes
0answers
125 views

Why is $M_{mn}(R)\simeq M_m(M_n(R))$? [duplicate]

Intuitively, it's not hard to believe that for a ring $R$, the matrix ring $M_{mn}(R)$ is isomorphic to $M_m(M_n(R))$. Taking a matrix in $M_{mn}(R)$ and turning the $n\times n$ blocks into single ...
1
vote
2answers
155 views

Resources for matrices and its applications

I was preparing some presentation slides on basics of matrices and its application. Even though, many of the participants are familiar with basic matrix operation, I planned to explain them by ...
3
votes
3answers
233 views

Rank and determinant of $D$ , an $n\times n$ real matrix, $n\ge 2$

Let $D$ be a $n\times n$ real matrix, $n\ge 2$. Which of the following is valid? $\det(D)=0\Rightarrow \mathrm{rank}(D)=0$ $\det(D)=1\Rightarrow \mathrm{rank}(D)\neq 1$ $\det(D)=1\Rightarrow ...
1
vote
2answers
123 views

Given a symmetric matrix $A$, are there any matrices $B$, $C$ that $BAC = I$?

Given a $4 \times 4$ symmetric matrix $A$, are there any matrices $B,C$ that: $BAC = I_{4}$ ? I've thought of $B$ being a orthogonal matrix $P$ ($B=P$) and $ C = P^{T}$ so we get $PAP^{T} = ...