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
2answers
43 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
1answer
73 views

Volume of a parallelepiped, given 8 vertices

Given the eight vertices $(0,0,0)$, $(3,0,0)$, $(0,5,1)$, $(3,5,1)$, $(2,0,5)$, $(5,0,5)$, $(2,5,6)$, and $(5,5,6)$, find the volume of the parallelepiped. I'm having trouble finding the 1 vertex ...
1
vote
1answer
20 views

How do I turn this Crank-Nicolson type equation into three vectors representing the middle, upper, and lower diagonals in a tridiagonal matrix?

I have the following homework problem: I have calculated the Crank-Nicolson equation to be Equation 1 $$ -200.05u_{m-1}^{n+1}+400.9995u_{m}^{n+1}-199.95u_{m+1}^{n+1} = ...
1
vote
1answer
33 views

What kind of matrix norm satisifies $\text {norm} (A*B)\leq \text {norm} (A)*\text {norm} (B)$ in which A is square?

$||A\times B||\le ||A||\cdot ||B||$ is not always correct. But which kind of matrix norm satisifies this formula for square matrix $A$ and arbitrary matrix $B$?
1
vote
1answer
702 views

Total unimodularity of matrix with consecutive ones property

A matrix has the consecutive ones property (often abbreviated C1P) if its every row (or column, for column-oriented C1P) is of the form $(0,\ldots,0,1,\ldots,1,0,\ldots,0)$. There is a theorem which ...
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 ...
0
votes
1answer
12 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
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
1answer
17 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 ...
0
votes
1answer
13 views

Relaxing the elements of a matrix

I try to understand a specific part of the paper "Consistent shape maps via semidefinite programming", where a binary symmetric Input matrix $X^{in}$ is given with $X^{in} \in \{0,1\}^{nm \times nm}$ ...
0
votes
1answer
1k views

Common coefficient matrix of two matrices

Consider the following two systems. (a) \begin{array}{ccc} 4 x - 2 y &=& -3 \\ x+ 5 y &=& 1 \end{array} (b) \begin{array}{ccc} 4 x - 2 y &=& 2 \\ x+ 5 y &=& 3 ...
-1
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 ...
27
votes
0answers
1k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
13
votes
0answers
236 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
13
votes
0answers
188 views

Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
11
votes
0answers
352 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $V= \begin{bmatrix} 1 ...
11
votes
0answers
484 views

Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 ...
8
votes
0answers
216 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
8
votes
0answers
166 views

Uniqueness of an infinite system of linear ODEs

How to prove that $\dot{x}=ax,\space x(0)=1$ has a unique solution if $a,x$ are infinite dimensional matrices? More specifically, let $Q$ be a bounded infinitesimal generator, i.e. ...
6
votes
0answers
78 views
+50

$5$ dimensional space over $\mathbb{R}$

When coming up with a double cover of $SO(5)$, I used conjugation by matrices of the form $$\begin{pmatrix} r & q\\ \overline{q} & r \end{pmatrix}$$ where $r\in\mathbb{R}$ and $q$ is a ...
3
votes
0answers
15 views

Optimal Matching Distance

I'm stuck on problem II.5.9 from Bhatia's Matrix Analysis. The problem is as follows: Let $\{\lambda_1,\dots,\lambda_n\},\{\mu_1,\dots,\mu_n\}$ by two $n$-tuples of complex numbers. Let $$ ...
2
votes
0answers
12 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 ...
1
vote
0answers
12 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
0answers
11 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
0answers
17 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
vote
0answers
11 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 ...
1
vote
0answers
23 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}}\ ...
1
vote
0answers
17 views

Bounding cosine of angle between vectors

Let $M$ be a symmetric, positive definite matrix such that $0\lt c_1 \le \lambda_{min}(M)\le\lambda_{max}(M)\le c_2$. I am trying to show that $\dfrac{v^TMv}{||Mv||||v||}\gt 0$ for $v\ne 0$ I ...
1
vote
0answers
14 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
12 views

Relationship between determinant and determinant of Hessian of determinant

Let $p = \det (x_{i}) \in \mathbb{Z}[x_i], i = 1, \ldots,n^2$ and $q = \det(H(p))$ the determinant of the Hessian of $p$. What is the relationship between $p$ and $q$. I have checked that $q = 2 ...
1
vote
0answers
30 views

Maximum of norm

Given a matrix $A$ with $N$ rows and $d$ columns, I would like to prove (or disprove) the following. Let $q(f)=\|(\begin{pmatrix} ...
1
vote
0answers
10 views

Taylor expansion of function of inverse matrix

Given the function $f(K)=K^{-1} ; domf= S^{n}_{++}$, i.e the set of positive definite matrices. So this is a convex function with respect of $K$. I want to find a Taylor expansion around $K_{0} \in ...
0
votes
0answers
8 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} ...
0
votes
0answers
17 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 ...
0
votes
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, ...
0
votes
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
0answers
12 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
0answers
11 views

Find the span of a set of matrices

I'm having significant difficulty with this problem: Let $M_{3\times3}$ be the set of all $3\times3$ matrices. Let $S$ be a subspace of $M_{3\times3}$ consisting of skew symmetric matrices. Show ...
0
votes
0answers
16 views

Are these matrix similar?

If two Laplacian matrices have same diagonal entries(say each diagonal element is $a$, in both matrices) and same spectrum. Are these matrices similar?
0
votes
0answers
10 views

transition and coordinate matricies

The problem statement, all variables and given/known data Consider the bases B = {b1,b2} and B' = {b'1,b'2} for R2, where b1=(1, -1), b2=(2,0), and b'1=(1,2), b'2=(1,-3) a. Find the transition ...
0
votes
0answers
28 views

Understanding notation - strange use of the del operator

I'm currently reading a paper with the following notation with the del operator which i have never encountered before: Does $\nabla _m$ just mean $\frac{\delta}{\delta \mathbf m} $ ? Furthermore, I ...
0
votes
0answers
14 views

Decomposition of a matrix into a product of two-level unitaries

I am working through example 4.12 and 4.13 from 'Quantum Computing: From Linear Algebra to Physical Realizations' (Pages 83-84 and 405-406) and I can't seem to figure out some what the '*' in figure ...
0
votes
0answers
48 views

when does the equality hold for the matrix norm triangle inequality and product inequality

So here is the problem: When does the equality hold for the following two famous "matrix" norm inequalities: $\|A+B\|\leq \|A\|+\|B\|$ $\|CD\|\leq\|C\|\|D\|$ For any norm you prefer. But I'm ...
-1
votes
0answers
54 views

invertible matrix and upper triangular matrix

If we are given a $A$ as $2\times2$ matrix. How to find an invertible matrix $P$ and a upper triangular matrix $U$ such that $A=PUP^{-1}$?
-1
votes
0answers
19 views

LQ Decomposition Dirty Paper Coding

Another question, what is the components of Q matrix hermitian from dirty paper coding? is it always has a square dimension? thank you
-1
votes
0answers
34 views

Find Homogeneous System from Solution Spaces

I have the following vectors in the Subspace of $\mathbb{R}^5$ $U=\mbox{Span}[(1,-1,-1,-2,0), (1,-2,-2,0,-3), (1,-1,-2,-2,1)]$ $W=\mbox{Span}[(1,-2,-3,0,-2), (1,-1,-3,2,-4), (1,-1,-2,2,-5)]$ I need ...
-1
votes
0answers
52 views

How to calculate discrete cosine transform for a matrix

I have a 8x8 matrix and I want to calculate its discrete cosine transform (DCT-II). I have this formula but I don't know to use it with a matrix. In the French Wikipedia they gave an example for ...
-1
votes
0answers
49 views

Is the eigenvalue decomposition equal to the singular value decomposition for real symmetric matrices?

Question is as the title states. I've read something similar for hermitian matrices, but am unsure if this is correct as well for real symmetric matrices.
-2
votes
0answers
14 views

Mathematics liner algebra

De fine what it means for two matrices A;B to be similar.?? and Are the matrices A = (1 2 , -1 1 ) and B = (5 3, -6 -3) similar to each other? Please justify your answer.