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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5 views

Diagonalization of 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 ...
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0answers
4 views

Markov chain ergodicity

$Xn$ 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 & 0 ...
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1answer
38 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
19 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 ...
0
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0answers
21 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 & ...
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2answers
26 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 \\ ...
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0answers
7 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 ...
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1answer
19 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
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1answer
9 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 ...
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0answers
14 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: ...
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0answers
14 views

Find a basis and the dimension of the eigenspaces of the matrix

Find a basis and the dimension of the eigenspaces of the matrix $$ \left( \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 3 & 0 \\ 2 & 0 & 1 \end{array} \right) $$ given that the ...
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1answer
10 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 & ...
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1answer
14 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 ...
1
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0answers
17 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 ...
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2answers
26 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 ...
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1answer
14 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 ...
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0answers
15 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.
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votes
4answers
382 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
votes
2answers
25 views

Prove Determinant [on hold]

A is the standard matrix of a counterclockwise rotation about the origin in R2 through an angle theta , show that det(A) = 1. How would one go about in solving/proving this? I know I have to use [ ...
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0answers
18 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 ...
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0answers
21 views

Mathematics liner algebra [on hold]

Define what it means for two matrices $A$ and $B$ to be similar. Are the matrices $$A = \pmatrix{1&2\\-1&1}$$ and $$B = \pmatrix{5&3\\-6&-3}$$ similar to each other? Please ...
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1answer
16 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) ...
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0answers
14 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 ...
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2answers
24 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} ...
10
votes
11answers
215 views

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

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 & ...
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0answers
21 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 &= ...
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votes
2answers
24 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- ...
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0answers
16 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 ...
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1answer
24 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
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0answers
13 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
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1answer
26 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 ...
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vote
1answer
25 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
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1answer
28 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
33 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
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2answers
14 views

On aquestion about nilpotent matrices

Let $A,B\in M_n(\mathbb C)$ be arbitrary matrices. 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 every $k\in \mathbb N$, can ...
1
vote
0answers
25 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}}\ ...
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votes
0answers
18 views

Why does an LU decomposition appear to exist for matrix A, when in reality, only a permuted LU decomposition exists?

I'm not quite sure how to word this question. I think it's best to show you my work instead. I need to find the $LU$ decomposition of matrix $A$ - which is straightforward. After all, if $A$ is a ...
2
votes
2answers
44 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 ...
0
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0answers
9 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, ...
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0answers
17 views

trace and determinant of subfield matrix?

Let L be a subfield of K, than it is : det_L (A) = det_K (A) det_L (A) = det_K (A) Now let us look at K as a vectorspace with dim(K)=d over L and the nxn matrix A over K firstly as a linear mapping f ...
0
votes
1answer
19 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
58 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
23 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
34 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
21 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
13 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 ...
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votes
1answer
35 views

Algebra - Solutions of linear systems

How would I find the real values of $k$ such that the following linear system does not have a unique solution? $$\begin{cases} x + 3y + kz = a \\ 2x + (2k+2)y + (3k-2)z = b \\ kx + (k+4)y + 4z = c ...
0
votes
1answer
13 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
13 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
16 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 ...