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
281 views

Inverting Matrix Equations Y = F(X)

I have a arbitrary $N \times N$ matrix $S$. I have an function of this matrix given by: $$A = F(S) = 2S + P^{-1}S + 2SP + PSP$$ where $P$ is the a cyclic permutation matrix which when acting on the ...
2
votes
2answers
607 views

Similarity between special matrices and special complex numbers

From Wikipedia: It is occasionally useful (but sometimes misleading) to think of the relationships of different kinds of normal matrices as analogous to the relationships between different ...
2
votes
1answer
111 views

Fast Algorithm For Adding An Equation To A System?

Assume an $N \times N$ matrix $A$ and a length $N$ vector $b$. I've already solved the system $Ax = b$ for $x$ using standard methods. (If you want you can assume that I have the inverse of $A$ as ...
2
votes
1answer
385 views

Wikipedia Article — Legendre Transform

I was reading the wiki article on Legendre Transform. I would be grateful if someone could explain the section at http://en.wikipedia.org/wiki/Legendre_transformation#Examples ie how they arrived at ...
2
votes
1answer
92 views

Looking for “average” of two permutations

I am a computer programmer and I am building a search engine for a client. Right now I am puzzling myself about the order in which I should return search results. There are two obvious orderings: ...
2
votes
2answers
506 views

Inverse of an upper-left triangular (partitioned) matrix

I'd appreciate help finding the inverse of the upper-left triangular (partitioned) matrix $$ \left[ \begin{array}{ll} \mathbf{K} & \mathbf{P} \\ ...
2
votes
2answers
762 views

Matrix Notation: What does A = [R | t] mean?

I am reading about homography in images and such. One thing pops up a lot: $\mathbf{P} = [\mathbf{R}|\mathbf{t}]$ What does this mean? Does this mean: If $\mathbf{R} = \begin{bmatrix}a & ...
2
votes
2answers
1k views

How to configure an LED to emit white light of a certain color temperature?

I'm working on an open source hardware project for a video/photo light, and it involves a fair bit of color math. I am trying to find my way from Color Temperature (CT) in Kelvin to current values ...
2
votes
2answers
697 views

Counting paths of a variable length on a directed graph

If I've been given a directed Graph $G = (V,E)$ and its adjacency matrix $A$. How do I calculate how many (directed) paths do exist from one specific node to another (specific) one? In general ...
2
votes
1answer
111 views

A question on the proof of $\text{pf}(MAM^T) = \text{det}(M) \cdot \text{pf}(A)$

There is a step in the proof of pf($MAM^T$) = det($M$) pf($A$) for a skew symmetric matrix $A$ that I do not understand. The proof I can find basically says that $$\text{det}(MAM^T) = (\text{det} ...
2
votes
1answer
88 views

Linear Algebra: An explanation on a simplification

Could someone please explain to me what property was used in simplifying this, or how this was achieved? Thank you.
2
votes
1answer
1k views

Solving a matrix equation $AX=XB$ in a CAS

I have the following computational problem. Let $N$ be a positive integer and $A\in \mathbb{C}^{2N\times 2N}$, $X\in \mathbb{C}^{2N\times 4}$ and $B\in \mathbb{C}^{4\times 4}$. I want to solve the ...
2
votes
2answers
2k views

How to 'minimize' correlation between series

Hi fellow mathemagicians, let's say that I have 3 series of numerical results (they represent 'drawdowns') : ...
2
votes
3answers
399 views

Is this a positive semi- definite matrix

I have a matrix $A$, which satisfies : $A$ is symmetric; all the diagonal entries of $A$ are equal to $1$; other entries of $A$ is between $0$ and $1$. My question is, whether $A$ is a positive ...
2
votes
1answer
884 views

Cross product for matrices

I have been contemplating extending the definition of cross product for matrices, and I wonder if this has been done before. Basically my definition is, given two 3x3 matrices: $A=(a_{ij})_{i,j=1} ^ ...
2
votes
1answer
383 views

Closed form for the exponential of a Lie algebra 3x3 matrix?

Matrices of the form: $$\begin{pmatrix} iz+iy&-iz+w&-iy-w\\ -iz-w&iz+ix&-ix+w\\ -iy+w&-ix-w&iy+ix\end{pmatrix}$$ where $x,y,z,w$ may be assumed to be real, form a Lie ...
2
votes
1answer
106 views

Does any equation holds for inverse(A- B)?

is it possible to calculate inverse of A-B, i have calculated inverse of A and B already. A and B are matrices
2
votes
1answer
162 views

expansion of an expression

The Fokker-Planck equation for several variables is : $$\frac{\partial W}{\partial t} = L_{FP}W\qquad(1)$$ where $$L_{FP} = -\frac{\partial}{\partial x_i}D_i(\{x\})+\frac{\partial^2}{\partial x_i ...
2
votes
1answer
454 views

Singular matrix

Suppose I have a singular matrix given by $$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{12} & a_{11} & a_{14} & a_{13}\\ a_{31} & a_{32} & a_{33} ...
2
votes
1answer
1k views

Gauss Jordan elimination - count of steps for $N \times M$ equation

I am having some problem wrapping my head around an assignment. I have to find out how many additions, subtractions, multiplications and divisions are used while solving an $N \times M$ linear ...
2
votes
1answer
27 views

Does $A-\lambda I$ have rank smaller than $A$?

Consider $\lambda$ as one eigenvalue of $A$, can we say that $A-\lambda I$ must have rank smaller than $A$? Or equivalently, $A-\lambda I$ spans a space which is a subset of $A$?
2
votes
2answers
38 views

A variation on the $AB$ vs $BA$ nonzero eigenvalues question.

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times m}$, so that $AB\in\mathbb{R}^{m\times m}$ and $BA\in\mathbb{R}^{n\times n}$ both exist. Thanks to Sylvester's determinant identity, we ...
2
votes
1answer
40 views

Group isomorphism between two groups .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. And consider $\mathbb{R}$ as an additive group. Prove that $SO_2(\mathbb{R}) \cong \begin{Bmatrix} ...
2
votes
1answer
27 views

For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?

Is it true that for every integer $n>1$ , there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $A \in M(n,\mathbb R)$ and $AD=DA \implies A$ is also a diagonal matrix ?
2
votes
1answer
57 views

Gaussian elimination algorithm performance

I am developing the quadratic sieve algorithm and I reached a new bottle neck: The matrix processing. I been reading quit a lot about this topic and I found many solutions Gaussian elimination: ...
2
votes
2answers
35 views

Finding Eigenvectors for $3 \times 3$ matrix with rows of zeros.

For a $3 \times 3$ matrix: $ $[A]$ = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} $ I have the eigenvalues: ...
2
votes
1answer
47 views

Condition number for computing $x$?

The question is: Consider the linear system $\left( {\begin{array}{*{20}{c}} 1&\alpha \\ a&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = ...
2
votes
1answer
28 views

How to Prove $V\otimes sl(k)=sl(V)$?

Let V be a vector space over a field $k$. Let $sl(n)$ be the set of all matrices elements from $k$ with trace zero. Is it true that $V\otimes _k sl(n)=sl(V)= \text{set of all $n\times$ n matrices ...
2
votes
1answer
67 views

Prove that $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent

Consider A to be a matrix that has all eigenvalues $\lambda$ with negative real part, that is, $Re(\lambda) < 0$. a)Show that the integral $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent, ...
2
votes
1answer
68 views

Proving that product of transpose matrix and the matrix is inversible

I need to prove that $A^T$$A$ is an invertible matrix. $$ A= \begin{bmatrix} \vec{a_1} & \vec{a_2} & \ldots & \vec{a_n} \\ \end{bmatrix} $$ Can I prove this using ...
2
votes
1answer
26 views

$A$ be $n×n$ matrix $A^{n}=0$ ,$A^{n-1}$ not equal to zero a vector $v$ belongs to R^n.then how to proof {V,AV,…A^(n-1)V} is a basis. [duplicate]

Given $A$ be $n×n$ matrix such that $A^{n}=0$, but $A^{n-1}$ not equal to zero a vector $v$ belongs to $\Bbb{R}^{n}$. Proof that {$V,AV,\cdots,A^{(n-1)}V$} is a basis.
2
votes
2answers
49 views

Random walk on a square

Problem: Given a square $ABCD$, $AB$ being an horizontal vertex, we start at $A$. With each step, we move to another corner: horizontally with a probability $p$ vertically with a probability $q$ to ...
2
votes
1answer
39 views

Eigenvalues of inverse matrix to a given matrix

How to calculate the eigenvalue of the inverse of a matrix given matrix is $A= \begin{bmatrix} 0&1&0\\ 0&0&1 \\4&-17&8\end{bmatrix}$ Is there any fast method?
2
votes
2answers
32 views

Proof of positive semi-definite matrix

Consider a matrix $X$ to be \begin{equation} X=P-PA^\top\left(APA^\top + Q\right)^{-1}AP, \end{equation} where $P\in\Re^n$ is a positive definite matrix, $A\in\Re^n$ is a non-singular matrix, ...
2
votes
1answer
46 views

Idempotent ideal matrices

Let $I$ be a nilpotent ideal in a ring $R$. It could be easily deduced, by the definition of product of ideals, that the full matrix ring $\mathbb M_n(I)$ for any natural number $n$ is also ...
2
votes
1answer
48 views

The determinant of a certain matrix

How to compute the following determinant? $$\left| \begin{matrix} 1 & x_0 & x_0^2 & \ldots & x_0^n \\ 1 & x_1 & x_1^2 & \ldots & x_1^n \\ \vdots & \vdots & ...
2
votes
1answer
18 views

Z modules spanned by row space of matrix invariant under matrix multiplication

I have met this strange looking problem on which I have no idea, from my course on Abstract Algebra dealing with modules: Let $ v_1,...,v_k \in \mathbb{Z}^n $ row vectors of length n over $ ...
2
votes
1answer
29 views

Is every “weakly square” matrix either a $0$ matrix, or a square matrix?

Call a matrix $A$ weakly square iff $\mathrm{det}(A^\top A) = \mathrm{det}(A A^\top)$. Then clearly, every square matrix is weakly square, and every zero matrix is weakly square. Question. Are ...
2
votes
3answers
62 views

determinant of pascal matrix- proof

Let $U_n$ be the upper triangular Pascal matrix, $L_n$ the lower triangular Pascal matrix of n-th degree, i.e. $$ u_{ij} = \begin{cases} \binom {j-1}{j-i} & \quad i\le j\\ 0 & ...
2
votes
2answers
61 views

If a matrix has a unique left inverse then does it necessarily have a unique right inverse (which is the same inverse)?

I know if a matrix has a left and right inverse then the inverses are the same and are (is) unique and the original matrix is a square matrix, thus if I have a matrix which has multiple left inverses ...
2
votes
1answer
27 views

$A>0$ , $\sum\limits_{i = 1}^n {{x_i}} = 1$ , $Ax = \rho (A)x$ ,can we say that $\rho (A) = \sum\limits_{i,j = 1}^n {{a_{ij}}{x_j}} $?

Let $A>0$(i.e, all $a_{ij}>0$). There is $x>0$(i.e, all $x_{i}>0$) such that: $\sum\limits_{i = 1}^n {{x_i}} = 1$ $Ax = \rho (A)x$ Can we say that $\rho (A) = \sum\limits_{i,j = ...
2
votes
1answer
21 views

If $Ax=c(x)e, \forall x$, then $A$ has rank one

How to prove if $Ax=c(x)e, \forall x$, then $A$ has rank one? $e$ is a vector with all entries one. $c(x)\in \mathbb{R}$, which is a constant depending on $x$ My method is by Gaussian ...
2
votes
1answer
41 views

The trace of the product of positive definite matrix with the inverse of itself plus another positive definite matrix

Let $A$ be an $n\times n$ positive definite matrix with eigenvalues $a_1,a_2,\dots,a_n$ in descending order. Let $T$ be an $n\times n$ positive definite matrix with eigenvalues in $t_1,\dots,t_n$ in ...
2
votes
2answers
28 views

Simultaneous diagonalization

Given two symmetric matrices $A,B\in\Bbb R^n$ how can we find if they are simultaneously diagonalizable? If they have such property how can we find $U$ such that $UAU'$ and $UBU'$ are simultaneously ...
2
votes
1answer
26 views

Diagonalizable transmit to submatrix

If $$\begin{pmatrix} A & B\\ \Large 0 & C \end{pmatrix}$$ is similar to a diagonal matrix, are $A$ and $C$ also similar to diagonal matrices?
2
votes
1answer
28 views

Finding the formula for a linear transformation given the transformation of the basis vectors.

Consider the basis $\{\vec{p},\vec{q}\}$ where $\vec{p}=(1,1)$ and $\vec{q}=(-1,0)$. Let $T:\mathbb{R}^2\to\mathbb{R}^2$ be the linear operator such that $T(\vec{p})=(1,-2)$ and ...
2
votes
1answer
45 views

Questions about invertible matrices…

If I have two $n \times n$ matrices $M,N$ that have an inverse, how do I show that: ${M^t}$ has an inverse with ${({M^t})^{ - 1}} = {({M^{ - 1}})^t}$ that if $M$ and $N$ are orthogonal (${M^{ - 1}} ...
2
votes
1answer
54 views

Finding the inverse of a “bow-shaped” matrix

Consider the matrix $$A = \begin{bmatrix} n_{+} & n_1 & n_2 & n_3 & \cdots & n_{r-1} \\ n_1 & n_1 & 0 & 0 & \cdots & 0 \\ n_2 & 0 & n_2 ...
2
votes
2answers
34 views

Proof about Diagonalization of A

The question asks WHY is it true that $$A^{n} = PD^{n}P^{-1}$$ I can never do proper proving in algebra; what I almost know for sure is that a proof by induction is the way to go here. But how do you ...
2
votes
1answer
32 views

Matrix ODE, defective eigenvalue: Where does the extra '$t$' come from?

Given $A \in \Bbb R^{2\times 2}$, the system $$ \dot X=AX $$ Has the solution $$ X= c_1e^{\lambda t}\xi_1+c_2e^{\lambda t}\left (\xi_1 t+\xi_2 \right) $$ Where $\xi_1$ is the unique ...