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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19 views

Square root of symmetric matrix

I have a symmetric matrix $A$. How do I compute a matrix $B$ such that $B^tB=A$ where $B^t$ is the transpose of $B$. I cannot figure out if this is at all related to the square root of $A$. How to ...
0
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1answer
28 views

Linear Transforms & Matrices

$T:R^4 -> R^3$ Linear Transform This matrix is $[T]_{B2}^{B1}$ = A =\begin{pmatrix}1&2&3&4\\1&4&0&2\\2&2&9&10\end{pmatrix} After elimination we get: ...
0
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0answers
18 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
1
vote
1answer
11 views

How to characterize elements in the Bruhat open cell?

This might be an elementary question. For simplicity, let's assume $G=GL(n,F)$, where $F$ is a local field. Let $U$ be the subgroup of upper triangular unipotents, $A$ the subgroup of diagonal ...
0
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1answer
17 views

Rotate $xyz$ by use of pitch and yaw around origin

I have a project for a game which uses pitch/yaw for the direction of a players head. The pitch ranges from $0$ to $180$ and the yaw is $0$ to $360$. Yaw modifies $X$ and $Z$, pitch modifies the $Y$, ...
0
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0answers
8 views

Finding dynamic range of rotation matrices

How do I theoretically calculate the maximum value the transformed output vector can reach after transforming a vector? If it is an eigen vector then the eigen value will tell the max scaling ...
1
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2answers
352 views

irreducible, diagonally dominant matrix

I am facing a problem for irreducible,diagonally dominant matrices. How to prove that irreducible, diagonally dominant matrix is invertible? Please help me in this problem.
4
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3answers
85 views

If A+tB is nilpotent for n+1 distinct values of t, then A and B are nilpotent.

Suppose A and B are $n\times n$ matrices over $\mathbb{R}$ such that for n+1 distinct $t \in \mathbb{R}$, the matrix A+tB is nilpotent. Prove that A and B are nilpotent. What I've tried so far: ...
0
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1answer
27 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
1
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2answers
79 views

Unsolvable(?) Assignment Problem

I've recently been trying to implement the Hungarian Method in C++, and I've been using 5x5 matrices to test my program. Last night I came across a matrix which neither I nor my program can solve. Is ...
0
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1answer
39 views

The geometric multiplicity

By given this matrix: \begin{pmatrix}0&a&0\\0&0&1\\0&0&0\end{pmatrix} Why for any a which is not 0 the geometric multiplicity = 1? and why for a = 0 the g.m. = 2? I don't ...
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1answer
19 views

Matrix Rank calculation

I have a matrix A . A can be written as A=B+D. I know rank of B. It is 3. Is it possible for A to have ranks $<3$ . If so please prove.
2
votes
1answer
221 views

Given two sets of vectors, how do I find a change of basis that will convert one set to another?

Given two sets of dimension $n$ vectors $\lbrace v_1 , v_2 , \ldots , v_m \rbrace$, $\lbrace u_1, u_2, \ldots , u_m \rbrace$, where $m > n$, is there a computational method (in particular, using ...
2
votes
0answers
23 views

Counting symmetric binary matrices with constant line-sum

I'm trying to count, as the title suggests, symmetric matrices with entries $0$ or $1$ and constant line-sum $k$. ($0 \leq k \leq n$). If you start listing the number of these on a table you'll get a ...
1
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2answers
17 views

Matricial differentiation $x x^{\top} b $

What is the drivative of $x x^{\top} b $ with respect to x, knowing that b is constant vector?
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7answers
17k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
0
votes
0answers
6 views

How to perform FFT-SPA decoding?

i am working with Fast Fourier Transforms based Sum Product Algorithm. Actually, I have to code it in MATLAB and Without using MATLAB library FUNCTIONS.. I have used the algorithm from Non Binary ...
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0answers
17 views

Matrix Theory: Orthogonal and Linear independent [on hold]

Suppose that we want to find a vector c orthogonal to both of a and b (a) Express the condition using two dot products. (b) Find any one vector c other than the zero vector with this property. (c) ...
1
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1answer
36 views

Rank of a Matrix Sum

I have matrices of $3\times3$ dimension such that S=A+B. I know there is one inequality connecting rank of the matrices A,B and its sum S? Could you write down that here. It will be a great help for ...
0
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1answer
17 views

Rank of a simple matrix series

Problem Specifications and Given conditions I have a matrix $L$ with rank 3 and dimension $ 3 \times 3$. $L = K_0+\sum_{n=1}^{\infty}K_i $ . Rank of $K_0$ is 3 and rank of L is also 3. Rank of ...
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0answers
16 views

Simple question about matrices cones

Fix a dimension $d>1$ here. A cone is non empty closed and convex subset $K$ of $\mathbb{R}^{d}$ such that $tv\in K$ whenever $t>0$ and $v\in K$ and $K\cap -K=\{0\}.$ I have a finite set of ...
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2answers
47 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
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0answers
46 views

How to prove the following rank problems [on hold]

I am quite confused with this question. How can I initiate and approach a solution. Thanks!
2
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4answers
570 views

Right invertible and left zero divisor in matrix rings over a commutative ring

If a ring $R$ is commutative, I don't understand why if $A, B \in R^{n \times n}$, $AB=1$ means that $BA=1$, i.e., $R^{n \times n}$ is Dedekind finite. Arguing with determinant seems to be wrong, ...
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2answers
104 views

How to find exponential of triangular matrix

I'm studying for an exam and I can't find this in my notes or in the book, but it's on a past exam... Given $A = \begin{bmatrix}-1 & 1\\0 & -1\end{bmatrix}$, $e^{tA} = \begin{bmatrix}e^{-t} ...
3
votes
2answers
59 views

Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
2
votes
0answers
42 views

What do you call the following operations on a symmetric matrix?

Suppose we have a symmetric matrix of the following form, where the diagonal is always zero: \begin{array}{cccc} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0\\ 0 ...
0
votes
1answer
29 views

What is the number of sign patterns in $\frac n2$ of columns (or rows) of Hadamard matrices?

Given a Hadamard matrix of size $n$, I want to know what is the number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same. My guess ...
6
votes
3answers
10k views

A matrix and its transpose have the same set of eigenvalues

Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma(A^T)$ where $A^T$ is the transposed matrix of $A$.
5
votes
1answer
46 views

Is S a group under matrix addition

Another matrix question! Let $$S=\{A \in M_2(\mathbb{R}):f(A)=0\}\text{ and }f\left(\begin{bmatrix}a&b\\c&d \end{bmatrix}\right)=b$$ Is S a group under matrix addition. Either prove that ...
2
votes
1answer
74 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
0
votes
1answer
23 views

How to judge if a symmetric matrix can be factorized into two vectors?

How can we judge if this matrix can be written as the product of a column and a row vector? $A=\begin{bmatrix} \alpha & a & \alpha\\ \beta & b & \beta\\ \alpha & a & ...
0
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0answers
97 views

Is there some fast and efficient way for solving $x$

Let $b$ be a given constant scalar between 0 and 1, and $A$ a given $N \times N$ transitional probability matrix (i.e., each row sum of $A$ is 1, and $0\le {A}_{(i,j)} \le 1$). Let $A\circ A$ denote ...
0
votes
0answers
32 views

How to further simplify this equation?

Given that V is an invertible $n$x$n$ matrix and $\Sigma$ is a diagonal rectangular $m$x$n$ matrix, U is an $m$x$m$ matrix, b is an $m$x1 matrix and $\lambda$ is a positive number, how do u further ...
0
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0answers
17 views

Spiral Matrix Procedure in Maple

I am very new to Maple and Math StackExchange. The last question I asked helped me very much so I thought I would try again. I am wanting to write a procedure to take a square matrix and have it ...
0
votes
0answers
12 views

Strassens Matrix Multiplication Algorithm to compute product of 2 4X4 Matrices

Im trying to learn starssens matrix multiplication Algorithm.So far i know that it uses 7 multiplications and replaces a multiplication by several additions and subtractions,to achieve better ...
0
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0answers
18 views

Mathematical Maze Generation

I have performed some research into maze generation through Java code and learned about different "perfect" maze generation algorithms here. I found good Java-based maze generation code here. I have ...
3
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0answers
600 views

Determinant of symmetric tridiagonal matrices

Given an $n\times n$ tridiagonal matrix $$A =\left(\begin{array}{ccccccc} ...
1
vote
1answer
36 views

Properties of Determinant of matrix sum/multiplication

!Hey there :) I am currently working on a topic in control engineering and I'm currently looking for some way to relate determinants of matrix combinations to the determinant of the elements. ...
8
votes
1answer
209 views

Möbius function from random number sequence

Consider some arbitrary number sequence like the decimal expansion of $\pi$ = {3, 1, 4, 1, 5, 9, 2}. Prepend the sequence with the number $1$ so that you get {1, 3, 1, 4, 1, 5, 9, 2}. Then plug it ...
2
votes
1answer
25 views

Column space of stochastic matrix.

Consider an arbitrary matrix $M \in \mathbb{R}^{n \times m}$. Denote the column space of $M$ as $\mathcal{C}(M)$. Is it always possible to construct a right stochastic matrix $S$ such that ...
2
votes
1answer
25 views

Why left multiplication when it comes to Markov chains?

When working with Markov chains and transition matrices $P$ we multiply from the left, meaning that for example $\mu^{(n)} = \mu^{(0)}P^n$ or that the stationary distribution satisfies $\pi = \pi P$. ...
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3answers
44 views

How to reverse matrix vector multiplication?

I'm using the simple matrix x vector multiplication below to calculate result. And now I wonder how can I calculate ...
1
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3answers
45 views

Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
2
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3answers
65 views

Real matrices with non-real eigenvalues

I know this covers a lot, so perhaps someone could redirect me to a helpful website. for a) I have no idea where to start on the proof, as I don't understand why this is true. for b) I also have ...
0
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2answers
37 views

$A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, show that $\text{rank}(AB)\le\text{rank}(A)$.

The problem is asking a proof for $\text{rank}(AB)$ is smaller or equal to $\text{rank}(A)$. Given the conditions $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Any idea about the ...
18
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2answers
2k views

necessary and sufficient condition for trivial kernel of a matrix over a commutative ring

In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
1
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1answer
20 views

matrix function onto and 1-1

I have just started a linear algebra paper and we are doing 1-1 and onto functions. I understand in theory what they mean, I just don't know how to prove them. For example: Define $f: ...
1
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1answer
20 views

Multiplication of diagonal matrices with identity

What would the result of this multiplication be, given that $A$ is an $m \times n$ rectangular diagonal matrix and $I$ is the identity matrix. $$A^TIA = \cdots$$
2
votes
1answer
53 views

Invertibility of $I-AB$ [duplicate]

I got a question in linear algebra: 1) Let A and B be $n\times n$ matrices. If $I - AB$ is an invertible matrix, then prove that $I - BA$ is invertible. Can someone tell me how to solve this ...