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)

4
votes
1answer
72 views

Markov chain ergodicity

$(X_n)_n$ is a discrete-time, time-homogenous Markov chain. I have have the following transition matrix and want to show whether the chain is ergodic. $$P = \begin{pmatrix} \frac{1}{2} & 0 & ...
1
vote
1answer
56 views

How to prove that $I+A^{T}A$ is invertible [duplicate]

Let $A$ be any $m\times n$ matrix and $I$ be the $n\times n$ identity. Prove that $I+A^{T}A$ is invertible.
3
votes
1answer
57 views

Prove matrices are of equal rank

Suppose $P$ and $Q$ are $n \times n$ matrices of real numbers such that $P^2 = P$, $Q^2=Q$ and $I-P-Q$ is invertible, where $I$ is the $n × n$ identity matrix. Show that $P$ and $Q$ have the same ...
1
vote
1answer
63 views

Bound for eigenvalues of some special matrix

Let $Tridiagonal(a, c, b)= \begin{vmatrix} c & b & 0 & \ldots & 0 \\ a & c & b & \ldots & 0 \\ 0 & a & c & \ldots & 0 \\ \vdots & \vdots & ...
1
vote
2answers
60 views

Determinant of a square matrix in a field [duplicate]

\begin{array}{rrrrr|r} b & a & a & \cdot \cdot \cdot & a \\ a & b & a & \cdot \cdot \cdot & a \\ a & a & b & \cdot \cdot \cdot & a \\ ...
1
vote
0answers
141 views

Coppersmith-Winograd algorithm

I'm interested in algorithms to compute matrix multiplications. Is the Coppersmith-Winograd algorithm similar to the Strassen algorithm ? I have two other questions: 1) Are the multiplications done ...
0
votes
1answer
43 views

Finding the matrix representation of a linear transformation $ T: P_{3} \to \text{M}_{2 \times 2} $.

Define a function $ T: P_{3} \to \text{M}_{2 \times 2} $ by $$ T \! \left( a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} \right) = \begin{pmatrix} a_{3} & a_{0} \\ a_{2} & a_{1} \end{pmatrix}. ...
0
votes
1answer
25 views

Eigenvector / eigenvalue pairs for a Markov Matrix

If we're given a $ \displaystyle 2 \times 2 $ Markov Matrix (so all entries are non-negative and columns add to 1) M$(a,b)$ such that $$M = M(a,b) := \begin{bmatrix}1 - a & b\\a & 1 - b ...
2
votes
1answer
335 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
0
votes
1answer
14 views

Transitions of matrix

$T: \Bbb R^3 \to \Bbb R^3$ and $S: \Bbb R^3 \to \Bbb R^4$ are matrix transformations whose standard matrices are $$T=\begin{bmatrix} 1 & 0 & 2 \\ 2 & 3 & 4 \\ 1 & 5 & ...
0
votes
1answer
68 views

Finding eigenpairs for Markov Matrices

If we're given a $ \displaystyle 2 \times 2 $ Markov Matrix (so all entries are non-negative and columns add to 1) M$(a,b)$ such that $$M = M(a,b) := \begin{bmatrix}1 - a & b\\a & 1 - b ...
2
votes
0answers
33 views

why is the sum over even and odd permutations the same?

let $m$ be an $n \times n$ matrix (over $\mathbb{R}$,say) and for a permutation $\sigma \in S_n$ define the monomial: $$ P_\sigma(M) = \prod_{j=1}^n m_{j,\sigma(j)} $$ let $\tau$ be an odd ...
1
vote
2answers
36 views

Find the eigenvectors of $ A = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix} $.

Find the eigenvectors of $$ A = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix}. $$ I know you can solve $ \det(A - \lambda I) = 0 $ to find the eigenvalues of $ A $, but I keep getting no free ...
0
votes
1answer
384 views

Finding a non-zero vector in both the column space and the null space of a nilpotent matrix

$A$ is a $ \displaystyle 10 \times 10 $ matrix such that $A^{3} = 0$ but $A^{2} \neq 0$ and therefore, by definition, $A$ is nilpotent. Is there a non-zero vector that lies in both the column space ...
0
votes
1answer
29 views

Solving a variable in a matrix equation?

I am having trouble solving for a in the problem below. I've simplified it down to: $e^{14} = ln(e^e \cdot a)$. I'm not really sure where to go from here.
10
votes
4answers
3k views

Why are eigenvalues of nilpotent matrices equal to zero?

If $A$ is a $ \displaystyle 10 \times 10 $ matrix such that $A^{3} = 0$ but $A^{2} \neq 0$ (so A is nilpotent) then I know that $A$ is not invertible, but why does at least one eigenvalue of $A$ ...
1
vote
0answers
69 views

Converting second order system into first order system (ODE)

The second order equation $\frac{d^2\vec{x}}{dt^2} = A\vec{x}\ + \vec{g}(t)$ models an earthquake's effect on a 7-story building. Let $x_j(t)$ be the displacement of the $j$th floor with respect to ...
1
vote
1answer
60 views

Standard Matrix transformations

Consider the matrix transformation T: R3-> R3 defined by T(x, y, z) = (–4x + 3y + z, 2x – 5y, 6y + 7z). Find the standard matrix for T, and use that matrix to find T(1, –3). my solution: T(X (row1) ...
1
vote
0answers
23 views

Changed matrix dimensions - now formula doesnt work. Related to computer science

I'm writing a computer code to solve a problem, and I ran into some difficulties. This is not a coding question, my problem is purely mathematical, I will explain. We have a matrix $M\in Mat(\mathbb ...
1
vote
2answers
54 views

Problem implementing a QR factorization

I'm trying to write a Fortran subroutine to compute a QR factorization using the Householder method. To test my routine, I compute the factorization of the following matrix: $$ A = \begin{pmatrix} ...
18
votes
11answers
544 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$? [duplicate]

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
1
vote
0answers
40 views

Solving a linear system in function of a parameter

Problem: Solve the following system in function of the parameter $b$: \begin{align*} \begin{cases} -bx + 2y - (2+b^2)z + bu &= -2 \\ x -2y + bz -u &= 0 \\ x + (2b-4)y + (2-b)z + (b-1)u &= ...
-1
votes
2answers
34 views

Converting a second order n x n system into a first order 2n x 2n system

Say I have the following second order 7 x 7 system of equations: $x_1'' = 10(x_2- x_1- 1)$ $x_2'' = 10(x_3- 2x_2+ x_1)$ $x_3'' = 10(x_4- 2x_3+ x_2)$ $x_4'' = 10(x_5- 2x_4+ x_3)$ $x_5'' = 10(x_6- ...
2
votes
1answer
56 views

Which matrices are covariances matrices?

Let $V$ be a matrix. What conditions should we require so that we can find a random vector $X = (X_1, \dots, X_n)$ so that $V = Var(X)$? Of course necessary conditions are: All the elements on ...
0
votes
1answer
84 views

Eigenvalue of a linear map (proof)

Let's assume that $V$ and $W$ are vector spaces over a field $\mathbb{K}$, $\lambda\in\mathbb{K}$, $\lambda\neq0$. $S: V\rightarrow W$ and $T: W\rightarrow V$ are linear maps. Prove, that $\lambda$ ...
1
vote
0answers
57 views

Is the Hankel Transform a Hankel Operator

The "Hankel Transform" is the infinite weighted sum of the Bessel function. At the top of the wikipedia article http://en.wikipedia.org/wiki/Hankel_transform it says Not to be confused with the ...
0
votes
1answer
35 views

column space of a matrix

If $A\in M_{m\times n}\mathbb{(R)}$, show that $\mathcal{R}(AA^t)=\mathcal{R}(A)$ and $\mathcal{R}(A^tA)=\mathcal{R}(A^t)$ where $\mathcal{R}$ denotes the column space of matrix. How can I prove it ...
2
votes
1answer
78 views

What Method is used for Projecting the Rauzy Fractal?

I am trying to construct the Rauzy Fractal (http://en.wikipedia.org/wiki/Rauzy_fractal), I have a Tribonacci word generator and have the stairs constructed but I can't seem to get the projection onto ...
1
vote
1answer
36 views

$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$ for all positive definite $A\in\mathbb{R}^{n\times n}$

Let $A\in\mathbb{R}^{n\times n}$ be positive definite and $v\in\mathbb{R}^n$. Let $\left\|\cdot\right\|_2$ be the Euclidean norm. Can we prove $$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$$ for ...
1
vote
1answer
53 views

When is the matrix of eigenvectors of a complex symmetric matrix orthogonal?

Given a complex symmetric matrix $A=A^\top$ with a matrix of eigenvectors $C$ (which have distinct non zero eigenvalues) it can be shown that $C^\top C=I$ and that $C^\top A C=D$ where $D$ is a ...
1
vote
2answers
40 views

On a question about nilpotent matrices

Let $A,B\in M_n(\mathbb C)$ be arbitrary matrices and $A,B$ isn't nilpotent. Prove that $A+sB$ can not be nilpotent for infinite $s\in \mathbb C$. I tried to show that: $trace((A+sB)^k)=0$ for ...
1
vote
0answers
59 views

How to get 2nd partial derivative of a function of two vector variables

I am having trouble to calculate the expression: $$ \textbf{C}_{\textbf{q s}}\ \dot{\textbf{q}}\ \dot{\textbf{s}} = \frac{\partial^2 \textbf{C}}{\partial \textbf{q} \partial \textbf{s}}\ ...
3
votes
2answers
78 views

Matrix Differential Equations

I am working on a practice problem with the following equation: $$ \frac{d^3 x}{dt^3} + (k + 1)\frac{d^2x}{dt^2} + (k+1)\frac{dx}{dt} + kx = 0 $$ I understand the first part which is to convert to a ...
2
votes
2answers
289 views

Invertible skew-symmetric matrix

I'm working on a proof right now, and the question asks about an invertible skew-symmetric matrix. How is that possible? Isn't the diagonal of a skew-symmetric matrix always $0$, making the ...
0
votes
1answer
70 views

“distance” metric between two bases modulo determinant, rotation and chirality

I'd like some kind of metric that tells me how similar two complete, not necessarily orthonormal bases (represented by non-singular matrices $B_1, B_2 \in \mathbb{R}^{n \times n}$) are to each other, ...
-1
votes
1answer
748 views

Stability of linear systems with complex eigenvalues

I'm very confused with stability of linear systems, especially when they have complex eigenvalues. Say I have the Matrix $\begin{bmatrix}-5 & 3\\-1&1\end{bmatrix}$. The eigenvalues are ...
6
votes
4answers
64 views

Proving that the elements of $A^n$ matrix are non-zero

Let $ A = \begin{pmatrix} 1 & 2 \\-1 & 1 \\ \end{pmatrix}$. Prove that for every positive integer $n$ there exist integers $x_{n},y_{n}$ such that $A^n= \begin{pmatrix} x_{n} & -2y_{n} \\ ...
0
votes
2answers
45 views

Finding Jordan form of a specific matrix

Let $A\in Mat_{3X3}(\mathbb{R})$ such that $A^2-2A+I=0$ and $A\neq I$. Find the Jordan form of $A$.
2
votes
3answers
59 views

Orthogonal matrix with determinant $-1$

I must provide a counterexample for the statement If $A$ is an orthogonal matrix, then $\det(A)=1$. I know that an orthogonal matrix may have a determinant of $-1$, but how can I find such a ...
0
votes
1answer
264 views

How do i scale 2D vector using matrix

I know that scale matrix is 2x2 { x, 0, 0, y } basis. My vector { 100, 2 } and i want to scale it using custom 2x2 matrix. I've read that if left operand is 2D row vector, then multiplying it on a ...
0
votes
1answer
32 views

Absorbing states and Irreducible sets

Question on the definition of Markov Chain matrices: Is it possible to have an absorbing state (i.e. a state where the probability of returning to itself is 1) within an irreducible set? I.e., if we ...
0
votes
1answer
14 views

Is the relation $\Psi: M_2(\mathbb{Z^*}) \rightarrow \mathbb{Q}$ a Function

Determine if the following is a function Let $\Psi: M_2(\mathbb{Z^*}) \rightarrow \mathbb{Q}$ by $\Psi\big( \left[\begin{smallmatrix} a&0\\ 0&b\end{smallmatrix}\right]\big) = \frac{a}{b}$ ...
0
votes
1answer
23 views

Do the Matrices representing Markov chains need to be square?

I assume so -- I ask in the context of defining an irreducible set. If a set is non-irreducible, you should be able to find a "smaller" Markov chain matrix nested within a larger one. That "smaller" ...
0
votes
2answers
61 views

Characterize a rotation matrix

Given a matrix $A\in M_{2 \times 2}(\mathbb R)$ or $M_{3\times 3}(\mathbb R)$ how to determine if it is a rotation matrix? Is there any theorem that characterize a rotation matrix just by looking at ...
1
vote
1answer
44 views

Decompose an invertible $4 \times 4$ real matrix into product of $4 \times 3$ and $3 \times 4$

If we have an invertible matrix $M$ that is $4 \times 4$ and $\left| M \right| \neq 0$ (i.e. it is invertible), is it possible to decompose it into two matrices $4 \times 3$ and $3 \times 4$ ...
0
votes
1answer
20 views

Diagonal factorization of upper triangluar matrix to unit uppper triangular matrix

How to compute Diagonal factorization of upper triangluar matrix to unit uppper triangular matrix. i.e U = D*M where U is upper triangular; D is diagonal; M is unit upper triangular.
0
votes
1answer
22 views

Under What Conditions Is $f:M\rightarrow \mathbb{C}$ Where M Is the Set of 2x2 Matrices a Function and Not a Function?

I came across a problem that I thought was interesting. I attempted to solve the problem below, and I would be grateful if someone would check my logic in what follows. Let the set M of all 2 by 2 ...
2
votes
2answers
74 views

Determinant of a otherwise constant matrix with a constant diagonal

$$\begin{vmatrix} a-E&-b&-b&-b&-b\\ -b&a-E&-b&-b&-b\\ -b&-b&a-E&-b&-b\\ -b&-b&-b&a-E&-b\\ -b&-b&-b&-b&a-E ...
1
vote
0answers
91 views

Gauss Jordan Elimination different answers

I have a question regarding Gauss Jordan Elimination. I have this matrix: \begin{bmatrix}2&1&5&0\\1&0&-3&1\\7&2&2&1\end{bmatrix} So at the start I will switch R1 ...
1
vote
0answers
15 views

Reduction of matrix $A$ to $B$ to find eigenvalues by Power method [duplicate]

How to reduce matrix $A$ to $B$ such that it has all eigenvalues and eigenvectors of $A$ but the dominant eigenvalue (eigenvalue with largest magnitude) is replace by $0$ ? I am using Power method to ...