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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-1
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2answers
47 views

Matrix with eigen values given find [closed]

Let$$ P=\begin{bmatrix} 0&-2&-3 \\ -1&1 &-1 \\ a&2 &b \end{bmatrix},$$ for some $a,b \in \mathbb{R}.$ Suppose that $1$ and $2$ are eigenvalues of $P$ and $$ ...
2
votes
1answer
29 views

Strange matrix multiplication behavior in Matlab

I noticed strange behavior of some calculations in Matlab. Matlab code listing: ...
0
votes
0answers
53 views

QR decomposition algorithm

According to G. W. Stewart (Matrix Algorithms: Volume 1, Basic Decompositions) given an $n\times p$ matrix $A$, let $m=\min\{n,p\}$. The Stewart's Householder triangularization algorithm (Chapter 4, ...
1
vote
3answers
132 views

Show that these matrices are congruent.

Let $K$ be a field of characteristic$\ne 2$ and $u$ be an invertible element of $K$. Show that $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ and $\begin{pmatrix}u&0\\0&-u\end{pmatrix}$ are ...
0
votes
0answers
15 views

Matching two configurations by minimizing angles between pairs of points

I want to match two point configurations by rotation. The configurations are given by two $m$ by $n$ matrices $\boldsymbol A$ and $\boldsymbol B$ with each row representing a point in $\mathbb{R}^n$. ...
0
votes
1answer
21 views

How do I calculate the Jacobian matrix of the transformation of a 1-m manifold to a chart (topology question)?

What I want to do is take a 1-m manifold (something like a circle), and transform a subset of that manifold into a chart. I want to represent that function from manifold to chart with a 1 x 1 matrix, ...
0
votes
2answers
69 views

Matrix exponential Differentiation

We have the equation $e^X = \sum_{k=0}^\infty{1 \over k!}X^k.$, where X is a matrix of dimension $3 \times 3$ . Now I have a function $f(x)=C_1x+C_2*\frac{x^2}{2} $ where $C_1,C_2,f(x)$ has ...
4
votes
1answer
48 views

Sum of Nonnegative Matrix and Diagonal Matrix

Setup: Let $D = D^T > 0$ be a positive definite and diagonal $n\times n$ matrix, and let $A = A^T \in \mathbb{R}^{n\times n}$ be nonnegative with zero diagonals. That is, $a_{ij} \geq 0$ for $i\neq ...
0
votes
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
22 views

Gaussian Elimination diagonal elements

For a matrix A, assume that B is the upper triangular matrix after applying Gaussian elimination on A. I want to calculate only the diagonal elements of the output matrix B in terms of $A_{ij}$ (the ...
0
votes
2answers
34 views

Matrices Problem

I am doing the Cambridge O/L 2012 M/J P1 4024/12 Paper, Question number 12 (b). $$m = \begin{pmatrix} 3 \\ -2 \end{pmatrix}, \quad n = \begin{pmatrix} -1 \\ 4 \end{pmatrix} $$ Given that $$sm+3n = ...
0
votes
1answer
11 views

Solution of an undetermined linear system under a constrain on the norm

Given an equation of the kind: $$\|\overrightarrow{y}-\hat{A}\overrightarrow{x}\|_2\lt\epsilon$$ in which the matrix $\hat{A}$ is given and is a matrix $N \times M$ with the number of columns greather ...
3
votes
3answers
223 views

Solving inhomogenous ODE

I have an inhomogenous ODE. The main issue here is variables are matrices. It is bit of matrix calculus. A solution would be highly appreciated interms of x . I guess we can use same methods for ...
0
votes
1answer
56 views

Proof wanted that there is no positive integer matrix with positive integer eigenvalues u,v,w, if $0<u<v$ and $1\le w-v\le 2$

I have the following conjecture : If u,v,w are integers with $0<u<v<w$, then there is a POSITIVE INTEGER 3x3 - matrix A with eigenvalues u,v,w if and only if $w-v\ge 3$. I approved the ...
1
vote
0answers
21 views

Possible known ways to improve Back Substitution

I am quite new to this field and just implement algorithms. I am currently using back-substitution as a way to invert a lower triangular matrix. I would like to ask if there are known ways to improve ...
3
votes
3answers
54 views

Geometric series of matrices

I am currently reading 'Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach' by J. Hubbard and B. Hubbard. In the first chapter, there is the proposition: Let A be a ...
2
votes
2answers
39 views

closeness of a set of vectors

Is there some measure that captures the "closeness" of a set of vectors? Say I have a matrix, $$ A = \left[ \begin{matrix} 0.8 & 0.15 & 0.05 \\ 0.82 & 0.09 & 0.09 \\ 0.78 & 0.08 ...
2
votes
1answer
18 views

Prove That For $Av=0$, $\alpha \in F$, $\alpha*v$ Is A Solution

Lets $v$ be a solution for $Ax=0$ prove that $\alpha*v$ is a solution too. Because in matrix's algebra $\alpha \in F$ can get out, $\alpha$*Av=A*$\alpha$v $\alpha$v is a solution too. is it right? ...
6
votes
1answer
59 views

Matrix restoring (modulo n)

Let $m,n\geqslant 2$, and $A\in \mathcal{M}_n(\mathbb Z)$ such that $\det A \equiv 1 \pmod m$. Does it (necessarily) exist $M\in \mathrm{GL}_n(\mathbb Z)$ such that $A\equiv M \pmod m$?
0
votes
2answers
18 views

what is the parametric function of the new Bezier curve?

The cubic Bezier curve can be given in matrix form as If a cubic Bezier curve is rotated by an angle 30 around x-axis what is the parametric function of the new Bezier curve?
0
votes
0answers
12 views

Lower bound on eigenvalues of class of matrices

I'm trying to get a lower bound on the smallest eigenvalue of these matrices. Let $\{\phi_i\}_{i=1}^{N+1}\subset R^N$ be unit norm vectors such that $\langle \phi_i, \phi_j\rangle=-\frac{1}{N}$ for ...
1
vote
0answers
28 views

Generator Matrix

I have a C in $F_2^6$ $(x_1,x_2,x_3,x_4) \to (x_1,x_2,x_3,x_4,x_1+x_2,x_3+x_4)$ for $x = (1,0,1,1)$ i get $c = (1,0,1,1,1,0)$ we know that $$c = G . x$$ G is the Generator Matrix in the solution ...
2
votes
1answer
37 views

Multiplication for matrices

Can I prove following theorem $$(\lambda A)*B =A(\lambda B)=\lambda (A*B)$$ like this? My proof of the first equal sign We know that an easier notation for a generell matris, e.g A, is $$A= \ ...
0
votes
2answers
51 views

Determinant of complex matrix

How is the determinant of a complex matrix calculated? Is it the same algorithm as for real matrices, but the determinant itself is complex instead of real? (I was unable to find any hints with ...
0
votes
1answer
29 views

How to apply Vandemonde determinant to show nilpotent here?

This is question 5.5 from Erdmann & Wildon (paraphrased): Consider 2 square matrices over complex number $A,B$. Let $C=AB-BA$. Assume that $AC=CA$ and $BC=CB$. (i) Show that $trace(C^{m})=0$ for ...
0
votes
2answers
62 views

How do you find a non zero vector in Linear Algebra?

The question is; The vectors $a_1 = (1, 1, 0)$ and $a_2 = (1, 1, 1)$ span a plane in $\Bbb R^3$. Find the projection matrix P onto the plane, and find a nonzero vector $b$ that is projected to zero. ...
0
votes
0answers
38 views

Determining whether a function $\delta: M_{3\times 3}(\Bbb F)\rightarrow \Bbb F$ is $3$-linear

I've been working some suggested problems for a class, and I can't seem to understand the proper way how approach these problems. I tried to follow the following example and apply it to another ...
2
votes
1answer
24 views

Relating the Error in Matrix Inversion to the Determinant

If $M$ and $\tilde M$ are invertible square matrices which are almost the same (you get to pick the norm) $$\tilde M-M<\epsilon$$ Then can we say that their inverses are almost the same (of ...
1
vote
1answer
59 views

Linear maps, matrix transformation

Suppose $T$ is an element of $L(P_3(\Bbb R), P_2(\Bbb R))$ is the differentiation map defined by $Tp = p'$. Find a basis of $P_3(\Bbb R)$ and a basis of $P_2(\Bbb R)$ such that the matrix of T with ...
0
votes
2answers
28 views

Matrix chain product

I’m Reading a book , and I’m stuck at a property of a product of matrix chain , it says that given $$A_{i..k }=A_{i}\times A_{i+1}\times A_{i+2} \times ...A_{k}$$ where every matrix is a ...
0
votes
1answer
22 views

Matrix Multiplication By Rows

Lets y be a row vector $(y_1 y_2... y_n)$ and A to be $nxm$ matrix \begin{array}{ccc} a_{11} & a_{12} & ... &a_{1m} \\ a_{21} & a_{11} & ... &a_{2m} \\ ... & ... & ...
2
votes
1answer
70 views

What causes commutativity of matrices?

My understanding is that the multiplication of two matrices is NOT commutative most of the time. One exception is two matrices, A and B, that are inverses of the other. This condition leads in turn, ...
1
vote
0answers
34 views

solution of infinite dimension linear sysmtem

Let $\{a_n\}_{n\ge0}$ and $\{b_n\}_{n\ge0}$ be decreasing sequences such that $a_0=A$, $\lim_{n\to\infty}a_n=0$ and $b_0=B$, $\lim_{n\to\infty}b_n=0$. For fix $n$, one can construct a $n$-dimensional ...
0
votes
0answers
22 views

Finding the least square fit for 3 parameters in Linear Algebra

I know how to find least square for $y = mx+b$ when we have two parameters. But this question has $3$ parameters, am trying to think of how to approach it but so far no success, I can't find any ...
1
vote
1answer
79 views

Given $A$, find invertible $B$ such that $B^{-1}AB$ is positive

Given $A \in Mat(n,n,\mathbb R)$, is there always an invertible matrix B, such that $B^{-1}AB$ is positive, assuming all eigenvalues of A are positive and simple ? If yes, is it possible to classify ...
1
vote
1answer
35 views

Finding a matrix projecting vectors onto column space

I can't find $P$, for vectors you can do $P = A(A^{T}A)^{-1}A^T$. But here its not working because matrices have dimensions that can't multiply or divide. help
0
votes
1answer
41 views

Faithful Representations of C*-algebras

Can anyone give me an example of a represetation of the algebra $M_n(\mathbb{C})$ that is not faithul? If it's not possible, could you explain me why it is not?
-1
votes
0answers
9 views

Is this matrix singular and of certain class of matrix?

2n*2n Matrix is given by $$ \pmatrix{ a_0&b_0&a_1&b_1\cdots & a_{n-1} & b_{n-1} &a_{n}&b_n\\ c_0&d_0&c_1&d_1\cdots & c_{n-1} & d_{n-1} ...
1
vote
1answer
12 views

Matrix translation by (1x2) vector

I'm having trouble figuring out how to approach this matrix translation question: Find the equation of the image line produced by translating all of the points on the line $y = 3x -1$ by the ...
0
votes
0answers
15 views

All permutation matrices that convert one Hadamard matrix into another Hadamard matrix.

Given a Hadamard matrix $H$, I know that applying row and column permutations, along with multiplying a row or a column with a -1 results in another Hadamard matrix $H^{'}$ equivalent to the first. ...
1
vote
1answer
25 views

Irreducible matrix equivalent connectedness of matrix graph?

If a matrix is irreducible, based on the following definition A matrix is reducible if there are two disjoint sets of indexes $I,J$ with $|I|=\mu$, $|J|=\nu$, $\mu+\nu=n$ such that for every ...
0
votes
1answer
21 views

How to construct orthogonal basis from a missing vectors?

I have $m$ vectors with a missing element each. $v_i=(*, a_{2i},\cdots,a_{ni})^\mathrm{T}\,\forall\, i\in\{1, \cdots, m\}.$ I would like to add the missing element $*$ to all $v_i$'s such that all ...
2
votes
1answer
38 views

Checking connectivity of adjacency matrix

What do you think is the most efficient algorithm for checking whether a graph represented by an adjacency matrix is connected? In my case I'm also given the weights of each edge. There is another ...
2
votes
2answers
24 views

Decomposing a square matrix into two non-square matrices

I have a matrix $A$ with dimensions $(mxm)$ and it is positive definite. I want to find the matrix $B$ with dimensions $(nxm), (n << m)$, which follows the following expression: $$A = B'B$$ Here ...
2
votes
2answers
147 views

How to prove these 2 matrix problems?

I'm reading a book and it gives that $\frac{\partial}{\partial A}Tr(AB)=B^T$, then it shows we can obtain $\frac{\partial}{\partial A}Tr(ABA^T)=A(B+B^T)$. But it seems we should have ...
0
votes
1answer
24 views

Getting linear combinations in linear algebra?

I failed a homework problem a few days ago. I can't figure out how they got the answers, which have been given in green as corrections. Help me figure how they got them;
0
votes
0answers
12 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
2
votes
1answer
36 views

Effective way of checking if all eigenvalues of a matrix are integers

Given A matrix with integer entries, it should be checked if all its eigenvalues are integers. Of course, the characteristic polynomial could be calculated, but is there any faster (or easier) ...
2
votes
0answers
32 views

Which n-tuples of positive integers can be the eigenvalues of some matrix with positive integer entries?

This question is closely related to some questions I already asked Given a tuple of positive integers (such as (1,2,5) ), is there a matrix A with positive integer entries such that the integers in ...
0
votes
1answer
42 views

Invertible Matrices Proof

Given that B is an invertible matrix and $B^3 + B^4 + B^7 = I$, find an expression for $B^{-1}$ in terms of only $B$. (where $I$ is an identity matrix) $B$ is a matrix that is $n \times n$.