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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1answer
183 views

Find a nonsingular matrix P given that A is similar to a Jordan matrix

Given ${\bf A}$ is similar to a Jordan matrix find a nonsingular matrix $\bf P$ such that ${\bf P}^{-1}{\bf AP}={\bf J}$ $$ {\bf A}= \begin{bmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 ...
0
votes
0answers
74 views

Criterion for coefficients of the inverse matrix

I need to determine the inverse of matrices of size $n \times n$. The result is only helpful if all coefficients in the inverse Matrix are strictly positive. Is there any criterion to the starting ...
0
votes
1answer
125 views

Find a Nonsingular matrix in Jordan Form

Let $$ A= \begin{pmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 1\\ \end {pmatrix} $$ Find a nonsingular matrix $P$ such that $P^{-1}AP$ is in Jordan form. The course I am taking ...
0
votes
2answers
713 views

What is the rotation axis and rotation angle of the composition of two rotation matrix in $\mathbb{R}^{3}$

I was told in class that a rotation matrix is defined by a rotation angle and rotation axis, if we call the rotation axis $v$ and take a basis of $\mathbb{R}^{3}=\{v\}\bigoplus\{v\}^{\perp}$ then the ...
0
votes
0answers
460 views

What is the BEST relaxation factor(ω) for Successive over-relaxation method?

I'm working on a project, implementing Successive over-relaxation (SOR) method (http://en.wikipedia.org/wiki/Successive_over-relaxation) using Python. SOR can only apply if given matrix is, ...
2
votes
2answers
262 views

Invariant subspace under orthogonal matrix

Let $V=\mathbb{R}^{n}$ and $T\,:V\to V$ be defined by $Tv=Av$ where $A\in M_{n}(\mathbb{R})$ is an orthogonal matrix. My lecture wrote that if $W\subset V$ is a subspace of $V$ then if $W$ is $A$ ...
2
votes
0answers
56 views

Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant $\mathbf{Question \;1}$. Let $$ A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right). $$ Then $C= \langle A\rangle$ ...
0
votes
1answer
287 views

A problem with definitions of rotation/reflection matrix/operator

I am a math undergraduate student taking a course called "Geometry and symmetry" and I have something I don't understand with the definition the lecture gave in class. Definition: ...
2
votes
1answer
74 views

Is there a special name for matrices consist of repeated unit vectors?

For example this one: $$Q=\begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 ...
1
vote
2answers
733 views

Integration problem in matrix calculus

Let $\mathbf{A}=\begin{bmatrix} f(x_1,x_1), & \ldots,& f(x_1,x_n)\\ \vdots&\ddots& \vdots \\f(x_n,x_1),&\ldots, &f(x_n,x_n) \end{bmatrix} $, where ...
0
votes
1answer
120 views

Covariance question: A non squared matrix possible?

I have an academic economic paper that says the following: $$q_r = \operatorname{Cov}(rx,v')\lambda$$ $$(14 \times 1)=(14 \times 4)(4 \times 1)$$ My a vector $q_r$ is of size $14 \times 1$ and my ...
0
votes
1answer
53 views

number of possible matrix entries swaps

Let us say that there is a $n \times n$ matrix with entries defined/given as some natural number. (duplicates are fine.) What would be the number of possible matrices that result from ...
1
vote
3answers
744 views

Find Elementary Matrics E1 and E2 such that $E_2E_1$A = I

I am studying Linear Algebra part-time and would like to know if anyone has advice on solving the following type of questions: Considering the matrix: $$A = \begin{bmatrix}1 & 0 & \\-5 & ...
1
vote
1answer
126 views

Description of a matrix in first-order logic

Assume a 9 by 9 matrix with variable elements that are natural numbers ranging from 1 to 9 (like a Sudoku puzzle). I want to describe the entire matrix in first-order logic, but I'm having trouble ...
1
vote
1answer
97 views

Please explain this notation equation

I am confused by this equation as I rarely use math in my job but need this for a program that I am working on. What exactly does the full expression mean? Note that $m^*_i{_j}$ refers to a matrix ...
1
vote
0answers
64 views

Can parallelism make faster matrix inversion algortihms? How?

My concern is about matrix inversion. Consider This page. I was thinking about creating four different threads for every component of the final matrix. In order to be more specific, I am going to ...
2
votes
1answer
1k views

2D transformation matrix to make a trapezoid out of a rectangle

In most vector graphic software libraries I can use (3x3) matrices to transform 2D geometry (e.g. scale, rotate, skew). How does a matrix need to look like to transform a 2D rectangle to a symmetrical ...
1
vote
1answer
474 views

Not all unitary matrices are orthogonal.

Recall that a matrix $A\in \mathbb{C}^{n\times n}$ is normal if $AA^{*}=A^{*}A$ where $A^*:=\bar{A}^T.$ Let $A\in \mathbb{R}^{n\times n}.$ Show that not all unitary matrices are orthogonal. Use ...
0
votes
1answer
63 views

Random samples from a normal distribution without explicitly constructing a covariance matrix

I have a set of samples in MxN matrix X that I want to use to approximate a covariance matrix: $$ C = X^\top X + eI $$ Where e is a small scalar and I is the NxN identity matrix. I need draw a ...
0
votes
3answers
174 views

Properties of a $3\times 3$ orthogonal matrix

Let $A$ be an $3\times 3$ orthogonal matrices with real entries,Then which are true $\det A$ is rational number $d(Ax,Ay)=d(x,y)$ for any two vector $x,y\in \mathbb{R}^3$ where $d$ is ussual ...
0
votes
0answers
78 views

Derivation of a vector equation

I am reading a derivation of conditional multivariate distribution, I have checked many time the following equation, I think it is wrong. $u$ and $v$ are any vectors, and $A$ is symmetric matrix. The ...
3
votes
0answers
129 views

Basis for completely bounded maps.

The set of completely bounded (CB) maps forms can be considered as a complex span of the set of completely positive (CP) maps. Can we find a basis for this complex linear space of CB maps such that ...
2
votes
1answer
118 views

How many $3\times 3$ binary matrices $X$ are there with determinant $0$ and $X^2=X^T$?

How many $3 \times 3$ binary matrices $X$ are there with determinant as $0$ that also satisfy $X^2 = X^T$?
0
votes
1answer
62 views

The Number of symmetric,PD, $8\times 8$ matrices

The Number of symmetric,Positive Definite, $8\times 8$ matrices having trace$=8$ and determinant$=1$ is $0$ $1$. $>1$ but finite. $\infty$ I am not able to do this one.
2
votes
1answer
48 views

Is there an explicit way to determine $\mathrm{Mat}_n(R[X_1,\dots,X_m])\simeq\mathrm{Mat}_n(R)[X_1,\dots,X_m]$?

For a commutative ring $R$, let $\mathrm{Mat}_n(R[X_1,\dots,X_m])$ denotes the matrix ring with entries from $R[X_1,\dots,X_m]$, and let $\mathrm{Mat}_n(R)[X_1,\dots,X_m]$ denotes the polynomial ring ...
4
votes
1answer
96 views

the solution of ode can be encoded as first order logic

Consider the following sytem of ODEs $\dot{x}= Ax$, and given $x(0)$, where $A$ is a $n\times n$ matrix with rational entries. Can I encode the solution, say $x(t)$ for a given $t$, as a first ...
0
votes
1answer
239 views

Calculate Upper Lower bound in $4\times 4$ correlation matrix

Let $X_i$, $i = 1,2,3 4$, be random variables on the same probability space such that $$\begin{align*} \mathrm{corr}(X_1,X_3) &= 0.3;\\ \mathrm{corr}(X_2,X_3) &= 0.1;\\ ...
2
votes
2answers
1k views

How to identify symmetric positive definite matrices?

I'm working on a project, implementing Successive over-relaxation (SOR) method (http://en.wikipedia.org/wiki/Successive_over-relaxation) using Python. SOR can only apply if given matrix is, ...
2
votes
1answer
95 views

logarithm of a matrix base a matrix — $\mathbf{A}^x = \mathbf{B}$

I want to solve $\mathbf{A}^x = \mathbf{B}$ where $\mathbf{A}$ and $\mathbf{B}$ are both $n$-by-$n$ matrices and $x$ is real. I see that in general there may be no solutions, or multiple solutions. I ...
7
votes
4answers
585 views

Cool/Useful Examples of Characteristic and Minimal Polynomials?

I'm teaching a Linear Algebra II undergrad course and for the section on characteristic & minimal polynomials, I really don't want to just give the students a bunch of matrices that have no ...
11
votes
1answer
403 views

spectral norm of random matrix

Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d entries with variance $\sigma^2/n$. What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$ ...
0
votes
1answer
1k views

“Change in rotation” matrix

Given two rotation matrices A and B (rotated from the same initial frame), how can I find the rotation matrix that represents the change in rotation from A to B? (I actually want to find the Euler ...
3
votes
3answers
180 views

the transformation which rotates a matrix by a half turn

Consider $$T_{2}: \left[ \begin{array}{cc} a & b \\ c & d \\ \end{array} \right] \rightarrow \left[ \begin{array}{cc} d & c \\ b & a \\ \end{array} \right] $$ $$T_{3}: \left[ ...
1
vote
1answer
197 views

Linear system with positive semi-definite matrix

I have linear system $Ax=b$, where $A$ is positive semi-definite, with positive elements and symmetric ($A$ is variance-covariance matrix ). I know that vector $b$ has elements $b_1>0$ and the rest ...
4
votes
3answers
120 views

Is there a standard operation to “rotate rings on matrices”?

Is there some standard operation to "rotate rings on matrices"? Look at the image below: The numbers around the four empty squares are what I'm calling ring, In the second matrix, this ring has ...
2
votes
1answer
632 views

Maximum likelihood covariance estimation of Gaussian

I was reading these notes on matrix calculus http://research.microsoft.com/en-us/um/people/minka/papers/matrix/minka-matrix.pdf and I could not figure out how to go from equation (30) to (31). Any ...
1
vote
1answer
82 views

Equivalent definitions on SVD property

Please have a look at the property of SVD - section "Singular values, singular vectors, and their relation to the SVD" from http://en.wikipedia.org/wiki/Singular_value_decomposition. Could it be ...
0
votes
1answer
73 views

Get the closed-form using jordan normal form?

I have got the following transition matrix: $$A = \begin{pmatrix} p & 1-p \\ 1-q & q \end{pmatrix}$$ How can one use the jordan normal form to get a closed-form to calculate such a values ...
0
votes
1answer
325 views

Multiplication of Matrix Properties - bi-symmetric, symmetric and anti-symmetric

Once again, can the great maths minds here please give a hand to explain or solve this problem. Much appreciated! How can you prove that a bi-symmetric matrix multiplied by symmetrical vector will ...
1
vote
3answers
126 views

Maximum numbers of underdetermined solution of $Ax=b$

Given $m \times n$ real matrix $A$, where $m<n$, we know that the nullity of $A$ is the dimension of the kernel $W=\{w| Aw=0\}$. Also all solutions of linear equation $Ax=b$ for $b\neq 0$ can be ...
1
vote
1answer
172 views

Rank of concatenated matrix

Given $N$ matrices $X_k=A_k^T\otimes (B_k\cdot C_0)$ and the vertically concatenated matrix $X = [X_1^T\; X_2^T\; \dots \;X_N^T]^T$, what is the condition for $X$ to have full rank? The matrices ...
5
votes
2answers
847 views

Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant

If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$? I remember seeing such a ...
2
votes
0answers
171 views

Characteristic function of matrix valued random variable

Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space and $X:\Omega\rightarrow\mathbb{R}$ a random variable. Denote by $\phi_X(u)$ the characteristic function of $X$. It is well known that if $X$ ...
6
votes
5answers
323 views

How to prove $I + t X$ is invertiable for small enough $ | t | ?$

Let $X \in \text{GL}_n(\mathbb{R})$ be an arbitrary real $n\times n$ matrix. How can we prove rigorously: $$ \underset{b>0} {\exists} : \underset{|t|\le b} {\forall} : \det (I + t X) \neq 0 $$ If ...
0
votes
0answers
181 views

Creating a precision matrix for Gaussian markov random field

I am currently trying to create a precision matrix for a Gaussian markov random field. Lets say I have random variables in a spatial grid of 6x6. Then I will have a precision matrix of 36x36. Now ...
2
votes
2answers
208 views

Computing determinant of a specific matrix.

How to calculate the determinant of $$ A=(a_{i,j})_{n \times n}=\left( \begin{array}{ccccc} a&b&b& \cdots & b\\ b& a& b& \cdots& b\\ \vdots& \vdots& \vdots& ...
3
votes
1answer
105 views

Is this vector derivative correct?

I want to comprehend the derivative of the cost function in linear regression involving Ridge regularization, the equation is: $$L^{\text{Ridge}}(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T\beta)^2 + ...
2
votes
2answers
833 views

Trace and eigenvalues under multiplication by a matrix with determinant $1$

Let $A,B$ be a $n \times n$ matrices such that $\det(B) = 1$. Will the spectrum (set of eigenvalues) $AB$ be same as that of $A$. Or, at least is $\mbox{Trace}(A) = \mbox{Trace}(AB)$ ? If not, what ...
12
votes
2answers
519 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...
2
votes
1answer
122 views

linear algebra: inverse of a matrix

The inverse of the matrix $A=\left( \matrix{1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} }\right)$ is ...