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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13 views

Convexity of the space of isospectral density matricies

Given the manifold $M$ of all complex, square matricies $\rho$ s.t. $\rho$ positive semidefinite $Tr(\rho)=1$ $\rho^{\dagger} = \rho$ consider the submanifold $N(\rho) = \{U\rho U^{\dagger} \ \ ...
-2
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3answers
146 views

How to find the terms in n th power of this matrix?

The Matrix Being : \begin{bmatrix}a&b\\b&-a\end{bmatrix} I need to find the terms in n th power of this matrix . I tried multiplying it for a few terms the pattern was hard for me to grasp ! ...
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0answers
24 views

Finding the kernel of these matrices

I am trying to find the kernel for each these matrices: \begin{equation} A = \left[ \begin{array}{ccc} 2 & 3 & 1 \end{array}\right] \qquad B = \left[ \begin{array}{c} 1 \\ 3 \\ -1 ...
1
vote
1answer
47 views

Transformation matrix determined by a basis

So I'm working trough a set of problems in preparation for a Linear Algerba exam, I'm stuck on this one: Let $\{x,y,z\}$ be the basis of $\mathbb{R}^3$ and $A:\mathbb{R}^3 \rightarrow\mathbb{R}^3$ ...
3
votes
2answers
59 views

A confusion about rotation group.

Definition: Let $G$ be a group and $S$ a set. Suppose to each group element $g$ there corresponds a function $f_g: S\to S$ in such a way that $f_g(f_h(s))=f_{gh}(s)$ for any $s\in S$ and any ...
1
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2answers
83 views

Rotating a set of anges (pitch/yaw/roll) by another set of angles (pitch/yaw/roll)

I want to rotate a set of angles (pitch/yaw/roll) by another set of angles (pitch/yaw/roll). By using Google I only found information about rotating a vector by angles, which is not what I need. ...
1
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1answer
42 views

Is matrix algebra a special case of Grassmann (exterior) algebra, and if so what is more general case?

Just a little question. I only recently heard about Grassmann algebra while reading a book on the history of vector algebra and quaternions. I still don't understand what does exterior product mean ...
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0answers
14 views

Relation between the cholesky upper triangular matrix of $A$ and the one of $A^{-1}$

Let $A$ be a symmetric, positive definite, real matrix (in short "spd matrix") and let $chol(A)$ be the upper triangular matrix obtained from $A$ by Cholesky decomposition. Is there any relation ...
4
votes
1answer
59 views

Number distinct root of $\det (P(\lambda )) = 0$

Let ${A_j} \in { \mathbb{C}^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a complex ...
3
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0answers
19 views

Equidimensionality of fibers of the map sending matrices to eigenvalues

If $M_{n\times n}(\mathbb{C})$ is the space of $n\times n$ matrices with coefficients in the complex numbers I can make a map: $\phi: M_{n\times n}(\mathbb{C}) \rightarrow Sym^n(\mathbb{C})$ which ...
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1answer
48 views

Square Matrix, diagonal matrix [duplicate]

Let $A = (a_{ij})_{n\times n}$ such that $AB = BA$ for every square matrix $B$ of order $n$. (i) Prove that $A$ is a diagonal matrix. (ii) Prove that $A$ is a scalar matrix. How to solve it? I ...
1
vote
2answers
68 views

Computing matrices to a power of $6$

Compute $\begin{pmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{pmatrix}^6.$ How would I solve this question. I found out that it's square would be $\begin{pmatrix} 2 & -2\sqrt{3} \\ ...
1
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2answers
57 views

Diagonalizable matrix over $\mathbb{C}$

I am asked to prove that a $3 \times 3$ complex matrix satisfying $A^3 = I$ can be diagonalized. I tried to study the eigenvalues of $A$, however this does not provide much information about if $A$ ...
2
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1answer
57 views

Existence of matrix of size NxN with following properties

You are given an integer $N$. ($N \le 100$). Does there exists a matrix $A$ of size $N\times N$ with the following properties? Each element of $A$ is either 1 or -1. The sum of products of ...
5
votes
2answers
80 views

Rotation matrix in $\mathbb{R}^2$. [closed]

Let $V = \mathbb{R}^2$ be two-dimensional Euclidean space, wit its usual $x$-coordinate and $y$-coordinate axes. Consider the linear transformation $L_\alpha: V \to V$ that performs a reflection about ...
1
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1answer
48 views

Expression for polynomial of companion matrix

I am rather stuck on an exercise concerning the companion/controllability matrix (the exercise stems from a course in control theory). Given the companion matrix \begin{equation} ...
9
votes
1answer
173 views

$n \times n$ matrices galore.

The group of $n \times n$ invertible matrices with entries in $\mathbb{R}$ is denoted $\text{GL}_n(\mathbb{R})$. Similarly, $\text{GL}_n(\mathbb{C})$ denotes the group of $n \times n$ invertible ...
0
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1answer
33 views

Inverse of the transformation $X \mapsto Y = X\cdot X^t$ [duplicate]

I have a Matrix Y of Kind : $n\cdot I$ where $I$ is the identity Matrix of size $n\times n$ , I need to find Matrix $X$ if it exists such that all elements are either 1 or -1 and satisfies $XX^t$ = ...
0
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2answers
65 views

Composition of Rotation matrix - how?

I am, at the moment, learning about rotation matrices, and it seems confusing to me how this is possible: $$R_A^C = R_A^B \cdot R_B^C$$ So.. $R_A^C$ must for a $2\times2$ matrix be defined as $[x^a ...
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2answers
36 views

How properties of a 2D hermitian matrix restrict the 2D matrix's elements

I have read different definitions, or properties, of a Hermitian matrix, and still am not sure if I have a sufficient number of properties to define a Hermitian matrix. Suppose the following is true ...
5
votes
1answer
93 views

If binomial expansion holds for $(A+B)^n$, does it follow that $A$ ad $B$ commute?

This is not a homework problem but my interest. Let $A, B$ a $2 \times 2$ matrix with all elements real. Is $AB=BA$ if $(A+B)^3=A^3+3A^2B+3AB^2+B^3$? Also, generalization for arbitrary positive ...
10
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3answers
125 views

If a real $2\times2$ matrix satisfies $A^4=I$, does it follow that $A^2=\pm I$?

I would like to know if a $2 \times 2$ matrix $A$ satisfies $A^4=I$, also satisfies $A^2=I$ or $A^2=-I$ if all elements of $A$ are real. Also, I would really appreciate your help on further ...
2
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0answers
52 views

Is the following inequality involving matrix exponential true?

Let $X$ and $L$ be real positive definite matrices. $$\operatorname{Trace}(X^{-1}(X - e^{\log(X) - L})^2) \leq \operatorname{Trace}(XL^2)$$ where the exponential and the log are matrix exponential ...
1
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0answers
92 views

On the Yale sparse matrix format

I am wondering if someone could provide me with additional information with regard to the so called Yale sparse matrix format, other that what can be already found here: ...
0
votes
1answer
61 views

Is this a valid norm?

For a positive semidefinite invertible matrix X define the following quantity over the set of symmetric matrices M $$f_X(M) = \sqrt{Trace(XM^2)}$$ Is $f_X(M)$ a valid norm? If yes is it easy to ...
0
votes
1answer
43 views

How to simplify $(aX+D)^{-1}$

Given a scale $a$, a full rank, symmetric, off-diagonal matrix $X$ and diagonal matrix $D$ Can the following inverse be simplified so that the scalar $a$ is not included inside the inverse operator? ...
1
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1answer
32 views

Let $\mathbf{A}=\mathbf{\hat{Q}\hat{R}}$. Prove that $\mathbf{Ax}=0 \iff \mathbf{Rx}=0$

Let $\mathbf{A}=\mathbf{\hat{Q}\hat{R}}$, where $\mathbf{\hat{Q}\hat{R}}$ is the reduced QR factorization of $\mathbf{A}$. Prove that $\mathbf{Ax}=0 \iff \mathbf{\hat{R}x}=0$ Intuitively, I know that ...
1
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1answer
36 views

Is $BAB'$ (with positive definite $A$ and full-row rank$(B) = k$) itself of rank $k$?

Here's the setup: A matrix $B$ with dimensions $k \times p$ with $k \leq p$ and rank$(B) = k$. A matrix $A$ with dimensions $p \times p$ is positive definite (not necessarily symmetric). Question: ...
1
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1answer
22 views

Is there a stochaistic matrix with an eigen value in the interval (0,1). any example will do.

Unless I did something very wrong, I don't think there should be an eigen value for a stochaistic matrix in that interval. Note, the stochaistic matrix doesn't even need to be regular, and the ...
0
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0answers
58 views

Fourier transform of a matrix

The Fourier transform of the matrix $\left(\begin{array}{ccccc} 0 & \alpha & 0 & 0 & \cdots\\ \beta & 0 & \alpha & 0 & \cdots\\ 0 & \beta & 0 & \alpha ...
0
votes
2answers
44 views

Why is $(\|x\|_2)^2$ given by $x^T\!x$ where x is a vector?

Suppose that $x$ is a real number, then $$(\|x\|_2)^2 = x \cdot x = x^2$$ Now suppose $x$ is a real vector, then $$(\|x\|_2)^2 = x^T\! x $$ Why should it be obvious that the multiplication sign ...
0
votes
1answer
23 views

Two row vectors and their determinant

I have two row vectors: [2 2] and [2 -2] and I need to compute determinant of these to see whether they are independent or dependent, i.e if determinant is not equal to zero they will be independent. ...
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0answers
39 views

Graph isomorphism in terms of permutation matrix elements

The graph isomorphism problem is defined as follows. If $\Gamma_1$ and $\Gamma_2$ are two graphs with adjacency matrices $A_1$ and $A_2$ respectively, is there a permutation $\pi$ such that ...
2
votes
1answer
140 views

Eigenvalues of real symmetric matrix [closed]

Suppose $A$ is a $n \times n$ real symmetric matrix with entries $a_{ij}\geq 1 $ and $a_{ii} = 0 $. Is it possible to have sum of the absolute eigenvalues of $A < 2 (n - 1).$
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0answers
32 views

Controllable and Observable system (Control Theory)

Allright, I have a system dx = Ax + Bu y = C*x I can get it in Canonical Observable form (in this form, it is NOT controllable), AND in Canonical Controllable form (in this form it is NOT ...
1
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1answer
82 views

Linear independence of a Matrix and its Transpose

I have a quick question. I got an $m\times n$ matrix $M$ of which I know that the columns are linearly independent. $M^T$ (its transpose) is then an $n\times m$ matrix with linearly independent ...
2
votes
1answer
29 views

Find the matrix A

Find the matrix A that has two rows and two columns and has $$A\pmatrix{1\\ 1}=\pmatrix{2\\ 1}\text{ and }A\pmatrix{-1\\1}=\pmatrix{1\\-1}.$$ Question: How do I write out the corresponding ...
0
votes
1answer
43 views

Creating a sparse matrix from a dense matrix

I would like to know whether there is a general method (and, if so, which one) to create a sparse matrix from a dense matrix. I know a sparse matrix simply does not include the zero entries, but since ...
1
vote
2answers
46 views

Choose a proper basis

An exponent $e^A$ of a diagonal matrix $A=\begin{pmatrix} \lambda_{1} & 0\\ 0 & \lambda_{2} \\ ...
1
vote
3answers
166 views

Find the $rank(AB)$ when rank of $A$ and $B$ are given

Let , $A_{7\times 5}$ be a matrix of rank $3$ and $B_{5\times 7}$ be a matrix of rank $5$. Then find the rank of the matrix $AB$. As we know , $rank(AB)\le\min\{rank(A),rank(B)\}$ , so $rank(AB)\le ...
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votes
1answer
19 views

Proving a matrix property of unnormalized laplacian matrix of a graph

Consider the symmetric real $n\times n$ matrix $W$. Define $d_i=\sum_{j=1}^{n}w_{ij}$. Define $D$ as the diagonal matrix with $d_i$ as its diagonal entries. Define $L=D-W$ as the laplacian matrix. Now ...
2
votes
1answer
60 views

To create a special matrix !!

How to create a $N \times N$ matrix with $1$ and $-1$ as its elements, such that when this matrix is multiplied with its transpose the resultant matrice is $N \times \mathbb{I}_N$. Where $N$ is a ...
0
votes
1answer
49 views

Block diagonal matrix multiplication

Given a matrix block diagonal matrix as follows ${\bf A} = \left[ \begin{array}{cccc} {\bf a}_1 & 0 & 0 & 0 & \\ 0 &{\bf a}_2 & 0 & 0 \\ 0 & 0 & {\bf a}_3& ...
1
vote
1answer
39 views

Given a unitary matrix, how do I show two of its components are unitary?

Consider a unitary block matrix $A$. $$A : = \begin{pmatrix}P &R \\ O & Q \end{pmatrix}.$$ Given that $P_{m,m}$, $Q_{n,n}$ how do I show that P and Q are unitary and R = O ?. where matrix ...
0
votes
0answers
44 views

Finite series for 4x4 matrix exponential of anti-Hermitian matrix

Based on the Cayley Hamilton theorem, one would expect an order 4 matrix polynomial formula for the matrix exponential of a 4x4 matrix. Is this polynomial known? Does it simplify in any way in the ...
0
votes
1answer
17 views

Method to find the set S of reals $λ$ such as $rg($M-I3)<3 given a matrix

Considering the endomorphism $f$ of $R^3$ of \begin{bmatrix} -3 & 5 & -5\\ -4 & 6 & -5\\ -4& 4 &-3 \end{bmatrix} relatively of the canonical base bc of $R^3$ find the ...
1
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1answer
30 views

Bounds on matrix norm of Hadamard powers

Suppose $A$ and $B$ are non-negative $n\times n$ matrices with $\|A\|= \|B\| = 1.$ In particular, therefore, $0\leq A_{ij}, B_{ij}\leq 1$ for all $i,j.$ Suppose also that $\|A-B\|\leq \delta.$ I ...
1
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0answers
25 views

Principal Component Analysis - first component product normalised to $1$

I am using $Matlab$ to run the $PCA$ of a set of data variables that I have. I have a $80 \times 13$ matrix, and have taken the first principal component using the following code: My question is ...
0
votes
2answers
55 views

Calculate the Eigenvalues of the 3 x 3 matrix

I have been given the homework question to determine the Eigenvalues of the following matrix: \begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1& 1 &1 \end{bmatrix} The ...
1
vote
0answers
31 views

Multi-dimensional Gauss integral with complex non-symmetric coefficients

Is there a closed formula that evaluates the integral, $$ I = \int_{-\infty}^{\infty} dx_1 .. \int_{-\infty}^{\infty} dx_D \exp\left(-\sum_{\mu=1}^D \sum_{\nu=1}^D a_{\mu\nu} x_{\mu} x_{\nu} ...