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
votes
2answers
215 views

FLOSS tool to visualize 2- and 3-space matrix transformations

I'm looking for a FLOSS application (Windows or Ubuntu but preferably both) that can help me visualize matrix transformations in 2- and 3-space. So I'd like to be able to enter a vector or matrix, ...
2
votes
0answers
17 views

Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
0
votes
1answer
31 views

Deriving equation in vector notation

I had some trouble deriving an equation from the book 'Elements of statistical Learning' p. 108 equation 4.9. This heavily relies on linear algebra, so I was wondering how the author came to his final ...
1
vote
1answer
130 views

how to Evaluate integral of density of Wishart matrix

Let $X_1 \cdots X_N$ are $N$ number of $m$ Dimensional Independent Complex Gaussian Random vectors Such that: $$ X_j \sim \mathcal{N}(\mu,\Sigma)\; \forall \;j=1 \cdots N$$ Let ...
1
vote
1answer
21 views

Finding the Formula of the Product of $e_{i,j}$ and $e_{k,l}$ to Return Zero Matrix

My teacher for calculus this year gave a handout on the first day with an excerpt from Rings, Fields, and Vector Spaces by B.A. Sethuraman. The reason for this is in the beginning of Sethuraman's book ...
0
votes
2answers
36 views

matrix powers problem

let $ A $ be the matrix :\begin{bmatrix}1 & 3 & 1\\4 & 2 & 3\\2 & 1 & 1 \end{bmatrix} Prove that $A$ verifies the expression : $ -A^{3}+4A^{2}+12A+5 I_{3} = O_{3}$ Deduct ...
3
votes
1answer
39 views

Matrix inequalities question

Let $A, B \in \mathbb{R}^{n \times n}$. Assume that: $$ 0 \preccurlyeq 2 A^\top A \preccurlyeq A^\top + A $$ $$ B^\top + B \preccurlyeq 0 $$ Is the following inequality true? $$ A B + B^\top ...
2
votes
2answers
89 views

Binomial Theorem on a Matrix

Does the expression follow binomial theorem? $(A + I)^n$ where $A$ is matrix, $I$ is identity matrix. I know the binomial theorem but do not know whether it is applicable to matrices also.
2
votes
1answer
288 views

Sorting Matrix to Block structure

I have a symmetric matrix and I want it to be as block-like as possible. I don't have a clear definition. I want the smallest number of groups of non zero elements or maybe the most non-zero elements ...
0
votes
0answers
8 views

CDF of smallest eigenvalue of non-central Wishart matrix - how to evaluate the integral.

Does anybody know how to derive the distribution of the smallest root of a non-central Wishart matrix? I have got an integral expression that would give me the desired answer but cannot solve the ...
3
votes
1answer
32 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
0
votes
2answers
26 views

Inverse of a Rotation matrix

If $R $ is a rotation matrix (determinant 1,orthonormal) can we say that $R^{-1}$ is also a rotation matrix? If yes how do we prove it?
1
vote
1answer
34 views

Permutation matrices

Let $\mathscr{M}$ be the set of all $n\times n$ matrices having entries $0$ and $I$ in such a way that there is one $I$ in each row and column. (a) If $M\in\mathscr{M}$, describe $AM$ in terms of ...
0
votes
0answers
7 views

What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
5
votes
2answers
93 views

An interesting linear algebra question

Let $A$ and $u$ be $n\times n$ matrix and $n\times 1$ vector of $\mathbb{C}$. Denote $\overline{A}$ is the matrix $(\overline{A})_{ij}=A_{ij}^*$, the conjugate number; ($\overline{A}$ is not the ...
0
votes
0answers
12 views

What are the upper bound and stability conditions for the following simple linear system

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
1
vote
0answers
40 views

How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
0
votes
0answers
13 views

Generalized Schur complement theorem

Let $M$ be an $(n+m)\times(n+m)$ real non-symmetric positive semidefinite (PSD) matrix partitioned as \begin{eqnarray*} M=\left(% \begin{array}{cc} A~~B\\ C~~D\\ \end{array}% \right), ...
0
votes
1answer
18 views

The nullity of a square matrix with linearly dependent rows is at least one. TRUE OR FALSE

Here is the answer my textbook gives. http://imgur.com/ycCRoWK I wonder: Why does the author ask this question specifically for square matrices? Is it different for other matrices.
0
votes
1answer
33 views

TRUE OR FALSE: Matrices with linearly independent row and column vectors are square.

Here is the answer of my textbook: http://imgur.com/vEoY31O Why must a matrice with linearly independent vectors have nullity(A)=0? That is where I lose track of the question. Are zero rows ...
2
votes
1answer
32 views

A characterization of a certain family of matrices in terms of another matrix.

Consider a real matrix $A$ of dimension $n \times n$. Assume $k \leq n$ is given. I am looking for ways to describe the following set of matrices in terms of properties of $A$. $\mathcal{S}(A) = \{B ...
2
votes
3answers
75 views

$A^2=cA$ for some $c \neq 0$

Let $A \in \mathbb{C}^{n \times n}$ and $0 \neq c \in \mathbb{C}$ a given constant. Suppose that $A$ has the following property: $$A^2 = cA.$$ Questions. 1) Is there a matrix class for matrices ...
0
votes
1answer
33 views

Normal Matrix Having all real eigen values is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not ...
1
vote
0answers
14 views

Leading eigenvalues of large sparse unsymmetric matrix

I have a matrix R which is sparse and all eigenvalues are -ve with a zero eigenvalue. Size of R is more than 1 million X 1 million. But I need to calculate only few large (by value not by magnitude) ...
1
vote
1answer
51 views

On Stochastic Matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
-1
votes
0answers
37 views
+50

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
1
vote
1answer
47 views

A solution of a linear system in some extension field implies a solution in the subfield

Fix a field extension $k\subseteq K$ and consider a linear system $Ax=b$ where $A$ is a matrix (not necessarily square) with coefficients in $k$. I don't understand why if the above linear system ...
0
votes
1answer
25 views

Prove that a normal matrix is unitary/Hermitian

I'm stuck with these two questions for while. I'd appreciate your help. ...
0
votes
2answers
41 views

How do you calculate this third eigenvector in this 3x3 matrix?

Scroll down to the bottom if you don't want to read how I arrived at my original two answers. My question is how are all the online calculators I check coming up with this third eigenvector (1, 1, ...
1
vote
0answers
16 views

Exponential and power of a special bidiagonal matrix

Given the bidiagonal matrix $$ \mathbf{A}=\begin{bmatrix} a_1 & b_1 & 0 & 0 & \dots & 0 & 0\\ 0 & a_2 & b_2 & 0 & \dots & 0 & 0 \\ 0 & 0 ...
0
votes
0answers
18 views

Immannt of a matrix.

I want to know in details about immanant of matrix. I have come to know about it in here but it could not give me sufficient knowledge. Kindly provide me some link where I can get introductory ...
0
votes
2answers
88 views

Determinants of Matrices det(4A) equals?

Suppose A is a 4 x 4 matrix such that det(A) = 1/64. What will det(4A^-1)^T be equal to? Here's my thinking, det(A^T) = det(A) I has no effect on the determinant. And det(A^-1) = 1/det(A) so ...
2
votes
3answers
1k views

Proving something is a square matrix

I don't want the solution. Please don't post the full solution. I just need a starting clue on how to do this. Suppose $A$ and $B$ are matrices such that $AB$ and $BA$ are defined. a) Show that ...
1
vote
1answer
20 views

Canonical form for orthogonal similarity classes

Could someone point me to a reference re canonical forms for classes of matrices in $M_n(\mathbb{C})$ which are unitarily similar? That is, canonical representatives for the equivalence class defined ...
0
votes
1answer
30 views

When does a matrix fail to be positive definite?

I am wondering how to think about a matrix being "bigger" than another. If I have the inequality X - Omega Sigma^-1 > 0 where all matrices are quadratic and X = Z'Z with Z positive definite and Omega ...
5
votes
2answers
8k views

Finding Transformation matrix between two 2D coordinate frames [Pixel Plane to World Coordinate Plane]

The question I'm trying to figure out states that I have N points (Pa1x,Pa1y) , (Pa2x,Pa2y)...(PaNx,PaNx) which correspond to a Pixel plane xy of a camera, and other N points (Pb1w,Pb1z) , ...
0
votes
0answers
9 views

Exponentially weighted rank ordered correlation matrix

Is there any well-known method to apply exponential weighting (similar to EWMA) to rank ordered correlation matrices such as Kendall tau or Spearman?
1
vote
0answers
12 views

Kronecker product and the vec operator: confusion on proof of vec(AXB) = (transpose(B) ⊗ A) vec(X)

I was reading up on Kronecker products and vec operator from couple of sources and landed on the equation: vec(AXB) = (transpose(B) ⊗ A) vec(X) suppose ...
0
votes
2answers
27 views

Show that every row of matrix $S$ is a linear combination of its bottom row and the row (1 1 1 1 1 1 )

Couldn't solve the following three questions. $$S=\begin{pmatrix} 36 & 35 & 34 &33&32&31 \\ 25 & 26 & 27&28&29&30 \\ ...
2
votes
0answers
36 views

What are the different solution concepts for Matrix-Ordinary Differential Equation [Theory Question]

I was recently given a ODE to solve from a boss at work, with the knowledge that I haven't done them before and this will help me learn. I've spent 10 hours so far learning the basics of ODEs. The ...
-1
votes
2answers
46 views

Conjugating by an upper triangular matrix does not change the diagonal entries.

This post http://mathoverflow.net/questions/49679/a-matrix-similarity-problem makes the claim that conjugating by an upper triangular matrix does not change the diagonal entries. But how do I prove ...
7
votes
1answer
90 views

Inverse matrix and its zero entries

Let $A$ be an $N \times N$ square invertible matrix with inverse $A^{-1}$. Is it possible to know through information of $A$ alone (i.e. without actually calculating $A^{-1}$) Which entries of ...
1
vote
3answers
44 views

Span of columns (or rows) of a given matrix?

Consider the following matrix: $$A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ \end{pmatrix}$$ The columns of $A$ span $\mathbb{R}^2$. The columns of $A$ span $\mathbb{R}^3$ ...
4
votes
1answer
48 views

An interesting property of symmetric real matrices with row and column sums zero

Let $A$ be an $n \times n$ real symmetric matrix with row and column sums zero. For example, $$ A=\begin{bmatrix}1 & -2 & 1\\ -2 & 1 & 1\\ 1 & 1 & -2 \end{bmatrix}. $$ I have ...
1
vote
1answer
59 views

Minimum eigenvalue of product of two matrices

Abstract description: Let $\mathbf{A}$ and $\mathbf{B}$ be two $n \times n$ real matrices. Let $\sigma( \mathbf{A B} )$ denote the spectrum of $\mathbf{A B}$. Assume that (A1) $\mathbf{A}$ is ...
12
votes
3answers
212 views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle ...
1
vote
2answers
86 views
+50

Lower and upper bound for the largest eigenvalue

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
3
votes
0answers
25 views

Analytical Matrix Inversion

I have a matrix of the form $A = bI - J$ where 1. $b$ is a large positive constant so that $A$ is positive definite 2. $J_{ij} = 0$ for $i=j$ and follows a power law off-diagnol. In index ...
1
vote
1answer
54 views

Determine cycle from adjacency matrix

Is there a way/algorithm to determine if there is a cycle in a graph if I only have the adjacency matrix and can not visualize the graph?
1
vote
0answers
23 views

Trick for Jordan-Matrix and transformation of basis

some time ago I found a 'trick' for getting a basis-transformation-matrix for jordan. I'd like to understand it, but at a certain point I stuck. Maybe you can help me? Given is a matrix A: ...