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), ...

learn more… | top users | synonyms (2)

2
votes
1answer
41 views

Weed out numerical artifacts from matrix inversion

I am working with the inverses to a set of large sparse matrices (in Matlab). A key indicator for my application is the number of non-zero entries in each row, and I recently discovered that I was ...
2
votes
1answer
49 views

How to decompose a matrix into an antisymmetric matrix plus a multiple of the identity

I was given a problem to solve earlier that I couldn't figure out. I don't still have it, but it was basically: Given the invertible matrix $A$, find the invertible matrix $P$, such that ...
2
votes
2answers
385 views

Find bases for eigenspaces of A

$$A = \begin{pmatrix} 6 & 4 \\ -3 & -1\end{pmatrix}$$ Find the bases for eigenspaces $E_{\lambda_1}$ and $E_{\lambda_2}$ of $A$. I don't really know where to start on this problem.
2
votes
2answers
208 views

Proof that this diagram commutes

This is an exercise in a book I'm reading: Define an $\mathbb R$-linear isomorphism $f_n : \mathbb C^n \to \mathbb R^{2n}$ by $$(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$$ ...
2
votes
1answer
25 views

One Linear Map a Polynomial in the Other: Help Find the Flaw in my Proof

$\DeclareMathOperator{\End}{End} \DeclareMathOperator{\spann}{span}$The following is a homework question for my Linear Algebra class that I did about a month ago, followed by my answer which only got ...
2
votes
1answer
50 views

Checking if a set of vectors are a basis

Find a basis for the subspace \begin{align} V=\{ (x_1,x_2,x_3,x_4)^T\in \mathbb{R}^4 | x_1-3x_2+5x_3 -6x_4=0\} \end{align} where $V\subset \mathbb{R}^4$ The basis ends up being spanned by the ...
2
votes
1answer
103 views

Cramers Rule. The why and how.

Can someone explain how Cramer's rule works. I understand the mechanics of it, and it's fairly straightforward to show algebraically that it's equivalent to GJ and substitution, but what's happening ...
2
votes
1answer
81 views

Counterexample for $\| (\Phi^\top \Xi \Phi)^\dagger \Phi^\top \Xi P \Phi \|_2 \leq 1$

All matrices are real. Let $P$ be any stochastic matrix (i.e. square, non-negative, rows summing to one). Let $\Xi$ be a diagonal matrix containing on its diagonal any left eigenvector of $P$ ...
2
votes
1answer
17 views

Every unitarily invariant matrix norm is sub-multiplicative?

Every unitarily invariant matrix norm is sub-multiplicative? In R. Bhatia, Matrix Analysis, after Proposition IV.2.4, it says that "Every unitarily invariant matrix norm is sub-multiplicative". But I ...
2
votes
1answer
100 views

Matrix of Shift Transform on arbitrary basis

This is problem 4.4.8 of Algebra by Artin. Let $V$ be a vector space with basis $(v_1,...,v_n)$ over a field $F$, and let $a_1,...,a_{n-1}$ be elements of $F$. Define a linear operator on $V$ by the ...
2
votes
4answers
57 views

Left inverse of matrix exists if $A\mathbf x=\mathbf b$ has a unique solution?

If the equation $A\mathbf{x} = \mathbf{b}$ has a unique solution for some $\mathbf{b}$ is it true that $A$ has a left inverse? $A$ is an $m\times n$ matrix.
2
votes
3answers
107 views

Homomorphisms into the General Linear Group

Let $G$ be a finite group of order $n \geq 2$. I want to prove that there always exists an injective homomorphism $\varphi:G \to GL_n(\mathbb R)$. Can you help?
2
votes
1answer
56 views

Linear Algebra: Symmetric matrices, diagonalization (help with proof)

I need a bit of help with an IFF proof, here it is: {Let X be a symmetric n × n-matrix. Show: $$X=Y^2$$ for some symmetric matrix Y iff X has only non-negative eigenvalues. } My thinking: This ...
2
votes
1answer
79 views

Calculating a Matrix Norm

I'm trying to calculate some norm for a matrix $A = [3, 2; 0,1]$ given the formula $\|A\| = \max_{|v|=1}|Av|$, where $|v|$ is taken to be the Euclidean norm for a vector, i.e. the standard distance ...
2
votes
1answer
17 views

Diagonalizability of an endomorphism on polynoms

Let $A,B\in\mathbb{R}[X]$ with $\deg B=n+1$. Let $\phi(P)$ be the remainder of the euclidian division of $AP$ by $B$. a) Show that $\phi$ is an endomorphism of $\mathbb{R}_n[X]$ (I have done that ...
2
votes
2answers
88 views

Find the $n^{th}$ power of a $2$x$2$ matrix.

Let $A=\begin{pmatrix}3&-2\\2&-2\end{pmatrix}$. Using Lagrange's interpolation compute $A^n$ for $n\in\mathbb{N} $ So far I've worked out the minimum polynomial of $A$ to be $(x-2)(x+1)$ but ...
2
votes
2answers
77 views

Proof that theorems about trace of matrix :

Can somebody help me about proofs of this theorems A is an nxn matrix and $\ A^2$ = mA then, tr(A) = m rank(A) . A is an nxn matrix and k is a positive integer then, tr($\ A^k$) = $\sum_{i=1}^n ...
2
votes
1answer
35 views

Get normalised eigenvectors

I am given the matrix: $\begin{pmatrix} a & b \\ b & -a \end{pmatrix}$ and I already calculated the eigenvalues $\lambda = \pm \sqrt{a^2+b^2}$. Now, I want to get the normalised ...
2
votes
2answers
2k views

Outer Product of Two Matrices?

How would I go about calculating the outer product of two matrices of 2 dimensions each? From what I can find, outer product seems to be the product of two vectors, $u$ and the transpose of another ...
2
votes
1answer
37 views

Linear Algebra - Invertible matrice

I have this problem and I'm not sure my solution is correct. Let $A$ be any $n \times n$ matrix, defined over the real numbers, A is not invertible matrix. Proof that there's is $B \neq 0$ and $C ...
2
votes
1answer
56 views

Parallelism in Golub & van Loan's Jacobi algorithm for symmetric eigenvalue problems

In Matrix Computations by Golub and Van Loan (3rd edition, page 433) an algorithm is given for a parallel version of the classical Jacobi algorithm for solving a real symmetric eigenvalue problem. The ...
2
votes
1answer
61 views

Matrix of a linear mapping

Consider $\mathbb{C}$ as a two-dimensional real vector space $\mathbb{R}^2$. Consider the linear map $z \to e^{i\theta}z$ on $\mathbb{C}$. What is the matrix of this map on $\mathbb{R}^2$ in the ...
2
votes
1answer
40 views

Matrix representation of the following equation - for finding optimal weights for regularized linear regression

If I have the following equation, $$E(w)=\sum_{i=1}^n (y_n -\beta^T x_n) +\lambda \sum_{i=1}^d \beta_i^2 $$ which is the cost function of regularized linear regression ($\beta$ and $x_n$ are ...
2
votes
1answer
28 views

Determinant of block matrix when $CD^T=DC^T$

When $CD^T=DC^T$ and $D$ is invertible we have: $$\left(\begin{array}{cc} A & B\\ C & D\\\end{array}\right)\times\left(\begin{array}{cc} D^T & 0\\ -C^T & ...
2
votes
2answers
126 views

Cholesky factor when adding a row and column to already factorized matrix

I have a positive deifnite, symmetrical, $N\times N$ real matrix $A$ which has 1's on the diagonal and all off-diagonal elements positive and $<1$. Let $A=LL^t$ be the Cholesky decomposition of ...
2
votes
1answer
99 views

What do I do with these equations to create a Jacobian matrix?

My instructions were to find equilibrium values (the picture I added is only showing E0, I was hoping if I got it figured out I could do the others rather than someone try to do all of them for me), ...
2
votes
1answer
80 views

Reversing a trimetric projection matrix

I am trying to determine exactly what the projection matrix is used by the game fallout 2. I am interested in making some similar projection. I found some information (ie, measures of the tiles etc) ...
2
votes
1answer
72 views

Generalized eigenvalues of overdetermined systems

I have a system of equations that can be written as ${(\bf{A}} + \lambda{\bf{B}}){\bf{x}} = 0$ Where ${\bf{A}}$ and ${\bf{B}}$ are $n \times m$, integer matrices. I know that there are several ...
2
votes
1answer
24 views

Solution for system of quadratic equations

Can anyone provide a straightforward solution to the following equation: $\vec{y}=M\vec{x}+N\vec{x^2}$ where $\vec{x^2}$ is a column vector with each component being the squared value of the ...
2
votes
1answer
66 views

expansion of matrix inverse

I would like to invert a square matrix $L$. One can write it as a sum of two matrices, one containing the diagonal terms ($D$) and the other the off-diagonal ones ($A$). $$L = D+A$$ I would like ...
2
votes
1answer
17 views

Can I write $[x_{i,j}]$ for the matrix whose $\{i,j\}$-th element is $x_{i,j}$?

Is it a general way to write $[x_{i,j}]$ for the matrix whose $\{i,j\}$-th element is $x_{i,j}$? Thanks.
2
votes
1answer
234 views

System (in a 6x6 matrix) of ordinary differential equations

One must find general solution for $$y' = \left(\begin{matrix} 1&2&-1&-2&1&2\\ -1&-2&1&2&-1&-2\\ 2&4&-2&-4&2&4\\ ...
2
votes
2answers
87 views

Are matrices vectors?

This may sound like an obvious question but it has confused me! According to wikipedia (https://en.m.wikipedia.org/wiki/Vector_(mathematics_and_physics)) vectors are defined as: "An element of a ...
2
votes
2answers
86 views

Eigenvalues of a special $M \times M$ matrix

I could not obtain an explicit formula for the eigenvalues of matrix $$ \begin{pmatrix} a & b & 0 & 0 & 0 & \cdots & 0 \\ c & a & b & 0 & 0 & \cdots & 0 ...
2
votes
1answer
37 views

Column/Row Space check

I have the following matrix: \begin{bmatrix} 1 & 2 & 0 & 1 & 0\\ 3 & 6& 1 & 6 & 1\\ 2 & 4 & -1 & -1 & -1\\ 4 & 8 & 0 & 4 ...
2
votes
1answer
70 views

A characterization of a certain family of matrices in terms of another matrix.

Consider a real matrix $A$ of dimension $n \times n$. Assume $k \leq n$ is given. I am looking for ways to describe the following set of matrices in terms of properties of $A$. $\mathcal{S}(A) = \{B ...
2
votes
1answer
80 views

Perron–Frobenius theorem

What is exactly the Perron–Frobenius theorem? In different books papers I read different statments, and I don't know what is the truth. In wikipedia there are also a lot of statements under this ...
2
votes
1answer
33 views

Solving a linear equation for a symmetric,positive matrix

Given the Problem $A x = b$ for some regular matrix $A \in \mathbb{R}^{n \times n}$ and $b\in\mathbb{R}^n$. One can compute $x$ with the Cholesky factorization in $O(n^3)$. If $A$ is known to be a ...
2
votes
1answer
44 views

Metric on the Set of Binary rectangular matrices

Consider a set of all possible Binary rectangular matrices. How many non-equivalent metrics can be defined? How to define non equivalent metrics on this set precisely?
2
votes
4answers
213 views

Independence of the columns of triangular matrices

Let $M$ be a square upper triangular matrix with nonzero diagonal entries. Prove that the columns of $M$ are linear independent. I understand that this proof can be done with some sort of ...
2
votes
1answer
140 views

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
2
votes
2answers
109 views

Prove that the determinants are equal

$$ Let\ A= \begin{bmatrix} 0 & a^2 & b^2 & c^2\\ a^2 & 0 & z^2 & y^2\\ b^2 & z^2 & 0 & x^2\\ c^2 & y^2 & x^2 & ...
2
votes
1answer
59 views

Is the trace of an idempotent matrix a sum of idempotents?

Let $R$ be a commutative ring, $n$ a positive integer, and $A$ an idempotent $n$ by $n$ matrix with entries in $R$. Is the trace of $A$ necessarily a sum of idempotents of $R$?
2
votes
1answer
216 views

Why is Doolittle Decomposition Algorithm failing and what should I try next?

I am trying to find the LU Decomposition of the following matrix: So far I have only tried the Doolittle Decomposition algorithm with partial pivoting (it's never failed me before!). As far as I can ...
2
votes
2answers
344 views

Solving for a Rotation Matrix Equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$

I would like to solve for an equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$ where $R_1$, $R_2$, and $R_\mathrm{x}$ are 3x3 rotation matrices, $R_1$ and $R_2$ are known, and $R_\mathrm{x}$ is unknown. ...
2
votes
1answer
43 views

Elementary Matrix and row of zeros

If you have the following matrix can $k$ be any number? \begin{pmatrix} 1 & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & 1 \end{pmatrix} So this is obviously an assignment question, ...
2
votes
1answer
179 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 $\pi = \pi P$. ...
2
votes
1answer
73 views

Basis for space of matrices in $\mathbb M_2(\mathbb R)$

Given that $G=\left\{ \left(\begin{array}{cc} a & -a\\ b & c \end{array}\right):a,b,c\in\mathbb{R}\right\} $ and $H=\left\{ \left(\begin{array}{cc} x & y\\ z & -z ...
2
votes
1answer
42 views

A generalization of GMRES

In oder to solve $Ax=b$, GMRES method finds $x_n$ in the $k$-th Krylov subspace i.e.: $$K_n=span\{b,Ab,...,A^{n-1}\}$$ and we have the condition: minimize $\|r_n\|_2$, which $r_n=b-Ax_n$ Now we ...
2
votes
1answer
42 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 ...