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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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 ...
8
votes
3answers
542 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
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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
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2answers
541 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
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1answer
504 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
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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
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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 ...
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1answer
317 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
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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 ...
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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
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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
179 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
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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
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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
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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} = ...
1
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1answer
248 views

Index notation clarification

Previously, I have seen matrix notation of the form $T_{ij}$ and all the indices have been in the form of subscripts, such that $T_{ij}x_j$ implies contraction over $j$. However, recently I saw ...
3
votes
2answers
624 views

Proof of existence of square root of unitary and symmetric matrix

I'm struggling with this exercise Let $U$ be a unitary and symmetric matrix ($U^T = U$ and $U^*U = I$). Prove that there exists a complex matrix $S$ such that: $S^2 = U$ $S$ is a unitary matrix ...
2
votes
1answer
266 views

How to make derivative operation in matrix space?

\begin{equation}\frac{d}{d\theta}\frac{1}{2}(\theta^TX - y)^2 = 0\end{equation} where, $X$ is $m $ on $ n$ matrix, $y$ is $m$-dimensional vector, $\theta$ is n-dimensional vector. I can solve this ...
4
votes
2answers
713 views

Partial derivative of trace of an inverse matrix

I have the following vector function $f(\mathbf{x})=\operatorname{Tr}[(\mathbf{A}+\operatorname{diag}(\mathbf{x}))^{-1}]$ where $\operatorname{diag}(\mathbf{x})$ is the diagonal matrix with values ...
0
votes
2answers
53 views

Is there a simple way of arriving at this solution?

Suppose we are given the matrix $$\begin{pmatrix}x'\\y'\end{pmatrix}=\begin{pmatrix}\cos(\omega t)& -\sin(\omega t)\\\sin(\omega t)& \cos(\omega ...
2
votes
0answers
109 views

Solution for this matrix equations (closed form or approximate solution)

Given a system of equations, I'm curious whether I can find the closed form solution for $P$, Here, $G$,$H$ are known $N \times N$ matrix, $\lambda$ is a known scalar; $s$,$t$ are two $N \times 1$ ...
2
votes
1answer
198 views

Find matrices $A,B$: exists $X,Y$: $AX-BY=[A,B]$ and $XA-YB=[A,B]$, here $X,Y$ are also matrices

Question Find matrices $A,B$: such that there exists matrices $X,Y$: $AX-BY=[A,B]$ and $XA-YB=[A,B]$, here $X,Y$ are also matrices, [A,B]=AB-BA - commutator. "Find" means "say something worth" any ...
5
votes
2answers
296 views

Special orthogonal matrix uniquely determined by $n-1 \times n-1$ entries?

For example, consider the specific question: Given $a_{11},a_{12},a_{21},a_{22}$ does that uniquely determine $A=\begin{bmatrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ ...
4
votes
2answers
271 views

$\det(A \otimes B - B \otimes A) = 0$ why? Why $rk(M) = n^2-n$ ? Why x and -x in Spec(M) ?

Let $A$, $B$ be $n\times n$ matrices. It seems $\det(A \otimes B - B \otimes A) = 0$. Moreover it seems that the kernel of $A \otimes B - B \otimes A$ contains $n$ vectors. Here is MatLab code to ...
0
votes
0answers
120 views

Matrix & Partition & Natural Number & Pattern

I would like to know if someone know, how is called a matrix M*N, where m represents the row index in the matrix and the sum of the N columns at this row. Meaning that each row represents a possible ...
0
votes
1answer
137 views

Is there any sense in zero-padding a matrix to make it $n\times n$ and find its eigenvalues?

I am debuging my Kalman filter and the Jacobian matrix of partial derivatives of h(measurement function) with respect to x(state) is not n×n, it is 13×16. $\displaystyle \quad\ \bf H_{[i,j]}$ = $\bf ...
1
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1answer
166 views

Linear Recurrence of Matrix into Closed Form Example Explanation?

I was given the following as an example of a linear recurrence and I don't understand how it works... Let us call the following eq1: $$x_i = \begin{bmatrix} \sum_{z = 1}^i{zk^{z-1}} \\ (i+1)k^i \\ ...
1
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1answer
331 views

Closed-form for eigenvectors of rotation matrix

For matrices that are elements of $SO(3)$ is there a formula for the eigenvectors corresponding to the eigenvalue $1$ in terms of the entries of the matrix?
3
votes
2answers
124 views

Counting matrices over $\mathbb{Z}/2\mathbb{Z}$ with conditions on rows and columns

I want to solve the following seemingly combinatorial problem, but I don't know where to start. How many matrices in $\mathrm{Mat}_{M,N}(\mathbb{Z}_2)$ are there such that the sum of entries in each ...
5
votes
1answer
506 views

Eigenvalues of $A+B$

$A,B$ are symmetric matrices, $A$ has eigenvalues in $[a,b]$ and $B$ has eigenvalues in $[c,d]$ then we need to show that eigenvalues of $A+B$ lie in $[a+c,b+d]$, I am really not getting where to ...
0
votes
1answer
173 views

Common eigenvectors of two special commuting matrices

Suppose you have a symmetric real 3x3 Matrix $S$ and an orthogonal matrix $O$ such that $O$ commutes with $S$, i.e. $OS = SO$. Suppose that $O$ is a nontrivial rotation about an axis in direction of ...
2
votes
1answer
1k views

Proof of 2 Matrix identities (Traces, Logs, Determinants)

I am working through a derivation in someone's thesis at the moment to understand an important result, but I am more than a bit rusty on matrices. Could anyone give me some tips on these identities? ...
1
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1answer
158 views

Cross product determinant's matrix

The cross product $a \times b$ can be represented by the determinant $$\mathbf{a}\times\mathbf{b}= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & ...
1
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0answers
224 views

Counting question on permutation matrices with rotation and imprinting

Please read question of distinct permutation matrices with rotation at first, then new counting questions are below: For a distinct $N\times N$ zero-symmetry permutation matrix, we could rotate it 3 ...
0
votes
2answers
152 views

Need help with finding matrix inverse in $\mathbb{Z}/26\mathbb{Z}$

I am trying to learn the Hill Cipher and I am facing difficulties understanding how to find the inverse of a matrix in Modulo 26. What I've learnt so far is that I need to apply elementary row ...
1
vote
1answer
63 views

$n\times n$ matrix with char poly $x^{n-2}(x^2-1)$

Let $A$ be an $n\times n, (n\ge2)$ matrix with char poly $x^{n-2}(x^2-1)$ Then which of The following is true? $A^n=A^{n-2}$, $r(A)=2$, $r(A)$ is atleast $2$, there exist non zero vector $x,y$ such ...
1
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1answer
158 views

Eigenvalues of a quasi-stochastic matrix

Quasi-stochastic In order not to make the title too long I used the term Quasi-Stochastic with this meaning: a quasi-stochastic matrix $Q$ is a square matrix $Q = ...
2
votes
2answers
217 views

Is it true that for matrices where all entry are lower than 1, determinant is lower than 1 as well?

Generic square matrix with positive 1 bounded entries Considering a matrix $A=(a_{i.j})$ where $0 \leq a_{i,j} < 1 \forall i,j$. It is important to consider that all entries are strictly lower ...
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votes
1answer
126 views

2-by-2 matrix determinant subrtraction

How to calculate determinant of this matrix? $\left[\begin{array}{cc} 1 & 2 \\ 0 & -2 \\ \end{array}\right]^3 . \left[\begin{array}{cc} 2 & 3 \\ -1 & 1 \\ ...
3
votes
3answers
212 views

Minimal polynomial and Hermitian matrix

A is a Hermitian $n\times n$ matrix over ${\Bbb C}$ . $$ A^m = I $$ for $m$ a natural number. Prove $$A^2=I $$ Suppose $m$ is odd. Prove $$ A=I$$ Well, for the first question I did this: Since $A$ ...
5
votes
0answers
57 views

What about other symmetric functions of the eigenvalues? [duplicate]

Possible Duplicate: Identities for other coefficients of the characteristic polynomial Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
0
votes
1answer
201 views

Constructing a matrix with given eigenvalues and given algebraic and geometric multiplicities

It's easy to construct a matrix with given eigenvalues, where the algebraic and geometric multiplicities agree by just picking random linearly independent vectors as the eigenvectors and then letting ...