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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0
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1answer
201 views

Prime number in a grid for an identity matrix??

I was reading this Transversal of Primes, and the solution shown for an 11x11 grid. Made me think of an identity matrix. First, have each $a_{ij}$ be either 1 for a prime number or 0 otherwise. ...
2
votes
5answers
3k views

Are the eigenvectors of a real symmetric matrix always an orthonormal basis without change?

I was reading the wikipedia page for symmetric matrices, and I noticed this part: ...
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 ...
2
votes
2answers
171 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 ...
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3answers
4k views

How to check if a matrix is positive definite

i want to know how to check if a matrix M is positive definite ,assume that M is 3x3 real numbers matrix i think one way is to put the matrix in a quadratic form $X^TMX$ , where X is a vector ...
3
votes
3answers
257 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
124 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
vote
1answer
334 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 ...
2
votes
1answer
330 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 ...
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
124 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
214 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 ...
6
votes
2answers
322 views

Fast multiplication of orthogonal matrices

Given $A,B\in SO(3)$, direct matrix multiplication computes $C=AB$ with 27 multiplies. The group $SO(3)$ is a $3$-dimensional manifold. This suggests that direct matrix multiplication, which thinks of ...
5
votes
2answers
327 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} \\ ...
1
vote
1answer
201 views

Generalization of the matrix concept

It has been some time since I left university... In a not too formal language, an $n$-dimensional vector is an indexed set of numbers $\{i_1, ..., i_n\}$. A $n\times m$ matrix is a set of numbers ...
4
votes
3answers
915 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 ...
1
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1answer
378 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
139 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 ...
0
votes
1answer
160 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 ...
5
votes
1answer
616 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
190 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 ...
1
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1answer
180 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 \\ ...
2
votes
1answer
2k 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? ...
2
votes
3answers
92 views

Show matrix polynomials are equal

Let $A$ be a matrix with no repeated eigenvalues: $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}.$ Let $p(x)$ and $r(x)$ be two polynomials satisfying $$p(\lambda_{i})=r(\lambda_{i}) \text{ for } ...
1
vote
0answers
250 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 ...
1
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2answers
2k views

Calculating formula to store location of Lower Triangular Matrix

I am struggling with a problem from this textbook. The question is as follows: Determine a formula h = f(i,j) to store location MATRIX[i][j] in h. Ensure to only store nonzero elements. Then it asks ...
0
votes
2answers
164 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 ...
5
votes
1answer
1k views

Eigenvalues of block matrices

Let $K$ be a field of characteristic 0, and consider the following block matrix $$M=\left(\begin{array}{cc} A & B\\ -B&D\end{array}\right),$$ where each block is an $n\times n$ matrix with ...
1
vote
1answer
179 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 & ...
0
votes
0answers
123 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 ...
1
vote
1answer
64 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 ...
3
votes
2answers
731 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 ...
1
vote
1answer
176 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
253 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 ...
2
votes
1answer
306 views

Exam Question: Find the dimension the subspace

I am putting an exam question of mine, which I hope complies with the overall policy of this forum. I have attempted it myself and I am checking effectively if my answer is right or not. EDIT: This ...
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votes
1answer
132 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 \\ ...
5
votes
0answers
61 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
226 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 ...
3
votes
3answers
239 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 $A$ ...
11
votes
5answers
324 views

Matrices with $A^3+B^3=C^3$

Problem: Find infinitely many triples of nonzero $3\times 3$ matrices $(A,B,C)$ over the nonnegative integers with $$A^3+B^3=C^3.$$ My proposed solution is in the answers.
0
votes
1answer
65 views

P-value not shown when there are too many variables in a linear regression

x<-c(1,2,3) y<-x^1.1+x summary(lm(y~x+I(x^1.1))) I have this code in R but it just is for the sake of easier understanding of what I am trying to achieve. ...
1
vote
2answers
636 views

eigenvalues by inspection

Well, May be my question does not make any sense, But one of my junior asked me whether we can say eigenvalues of a matrix by inspection may be for $3\times 3$ matrix? He said for the following ...
2
votes
2answers
999 views

What 's the short proof that for square matrices $AB = I$ implies $BA = I$? [duplicate]

Possible Duplicate: If $AB = I$ then $BA = I$ I'm trying to remember the one line proof that for square matrices $AB = I$ implies $BA = I$. I think it uses only elementary matrix properties ...
2
votes
3answers
6k views

Inverse of transformation matrix

I am preparing for a computer 3D graphics test and have a sample question which I am unable to solve. The question is as follows: For the following 3D transfromation matrix M, find its inverse. Note ...
1
vote
1answer
78 views

Atypical equation over matrices

Let $M$ be a $n\times n$ matrix over $GF(2)$. Is there any standard algorithm to solve equations of the form $M^{n-1}v_0 + M^{n-2}v_1+...+M^0v_{n-1}=const$ where $v_i$ are vectors with exactly one ...
2
votes
1answer
519 views

Decomposition of a symmetric matrix into multiplication of two vectors

What is the necessary condition for a real symmetric matrix $ A_{m\times m} $ to be written as $B*B^T$ where $B$ is an $(m\times 1)$ matrix ?
0
votes
1answer
76 views

The matix of $A^*$ (Banach space adjoint of $A$) with respect to the dual basis coincides with $A^t$ (the transpose of $A$).

Let $A\in B(\mathbb{C}^n) \cong \mathbb{M}_n(\mathbb{C} )$ Prove: The matix of $A^*$ (Banach space adjoint of $A$) with respect to the dual basis coincides with $A^t$ (the transpose of $A$).
4
votes
1answer
99 views

Is this function convex when the input vector is positive?

I am wondering if $f(\mathbf{x})$ is convex on the input of a vector of $n$ positive reals $\mathbf{x}$: $$f(\mathbf{x})=\operatorname{Tr}[(\mathbf{A}+\operatorname{diag}(\mathbf{x}))^{-1}]$$ where ...
0
votes
0answers
138 views

a minimum and maximum value problem

First and foremost, I greatly appreciated the prior attempts made by the excellent mathematicians Robert Israel, and mixedmath on the related problem. Now I have the following problems of the ...
3
votes
2answers
877 views

How to find a 2D basis within a 3D plane - direct matrix method?

I have a plane equation in 3D, in the form $Ax+By+Cz+D=0$ (or equivalently, $\textbf{x}\cdot\textbf{n} = \textbf{a}\cdot\textbf{n}$), where $\textbf{n}=\left[A\:B\:C\right]^T$ is the plane normal, and ...