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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1answer
51 views

A is a Hermitian projection if and only if it is an orthogonal projection

I need to figure out this property of Hermitian / Orthogonal projections "A is a Hermitian projection if and only if it is an orthogonal projection" Your assistance will be highly appreciated. ...
0
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1answer
26 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
1
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1answer
29 views

c0mpatible system $A^TAx=A^Tb$

Let $A\in\mathbb{R}^{n\times n}$ be a singular matrix. Prove that the system $$A^TAx=A^Tb$$ is compatible for any $b\in\mathbb{R}^n$. I want to prove that $A^Tb\in Ran(A^TA)$,i.e. $A^Tb\bot ...
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0answers
17 views

minimal residual method converge for SPD matrix

Prove that the minimal residual method($D=H=I$) converges for any matrix $A$ which is positive definite in $\mathbb{R}^n$
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0answers
25 views

Proving that $\mathrm{rank}(P_1+P_2) = \mathrm{rank}(P_1)+\mathrm{rank}(P_2)$

Supposing $P_1$ and $P_2$ two projectors as: $P_1\circ P_2 = P_2\circ P_1$. What is the condition for $P_1+P_2$ to be a projection? If it was the case above then how can I prove that ...
2
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2answers
54 views

Showing that matrix admits an eigenvector?

Let A= a b c d be a 2 x 2 matrix, where a,b,c and d are real numbers. We say that A admits an eigenvector if there exists a unit vector u and a real ...
0
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2answers
30 views

How to find the left and right eigenvectors of a matrix corresponding to a zero eigenvalue

Let $$A=\begin{pmatrix}1&3&-2\\1&-8&8\\3&-2&4\end{pmatrix}$$ Find non-zero vectors $u$ and $v$ satisfying $Au=0=A^Tv$
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1answer
24 views

Generate some random matrix with given rank

Very often for creating new exercises (I teach basic matrix algebra), I need to a find a matrix $A$ such that: it has integer coefficients, not too big (in order to avoid big numbers computations) ...
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2answers
57 views

When do eigenvectors converge?

Let $A_n$ be a sequence of self-adjoint $N\times N$ matrices that converge in the operator norm to $A$. The sequence of eigenvalues of $A_n$, denoted $\lambda_n$, converges to an eigenvalue of $A$, ...
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0answers
57 views

Shortened Generator Matrix

goodmorning, could someone tell me if the following code has been handled correctly? I have this generator matrix (which I should modify in order to have it correct): $$G=\begin {bmatrix} ...
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0answers
12 views

When does the leading right eigenvector gives the stationary distribution?

I am trying to make sense of the meaning of the leading right eigenvector in mathematica modeling (applied mathematics). I am interesting in models of the kind $\overrightarrow v(n+1) = M ...
5
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2answers
58 views

Exact meaning of “Not every matrix is a tensor”.

I've recently begun reading about tensors and am trying to understand the second order variety in the context of euclidean $\mathbb{R}^n$ with orthonormal basis {$e_1, e_2,\ldots, e_n$}. This seems ...
1
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3answers
65 views

Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? [duplicate]

Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
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0answers
18 views

reducibility of an operator

This is perhaps a very basic problem in Linear Algebra, which concerns the ability to reduce an operator. An operator $A$ on a finite dimensional vector space $V$ is called reducible if there exists ...
1
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0answers
23 views

Finding mathematical relation of matrices with reverse indices

I am designing a simple game, I have faced this problem to get the mathematical relation between two kind of tables: MATRIX A MATRIX B As you can see the table A (or Matrix A) is the normal ...
2
votes
1answer
29 views

How to get an eigenvector of a $3\times 3$ matrix that has first column and a row of zeros

I have the following matrix $$ \begin{bmatrix} 1& 0& 0\\ 0& 1& 1\\ 0& 1& 1 \end{bmatrix} $$ First I got the eigenvalues which are $0$, $1$, $2$. I tried to get the ...
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0answers
30 views

On Matrix polar decomposition and absolute value operator

The polar decomposition for complex matrices is $A=OP$ where $O$ is a partial isometry and $P$ is (hermitian) positive semidefinite. In other notations the matrix P is considered as an absolute value ...
2
votes
1answer
60 views

Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
0
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0answers
24 views

p.d.f and distribution of multivariate normally distributed variables

Suppose $X\sim N(\mu,V)$ where $\mu = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$ $V = \begin{pmatrix} 3 & 2 & 1 \\ 2& 4 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ a) ...
0
votes
0answers
18 views

Identifying the abelian group with a presentation matrix

I am doing problems from Artin: \begin{bmatrix} 2 \\ 1\\ \end{bmatrix} and \begin{bmatrix} 2 & 4\\ 1 & 4\\ \end{bmatrix} For the First one after manipulating rows I ...
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votes
0answers
8 views

Inversion of $H=L'W'WL$

Given $L$ is a square and ful-rank matrix and $W$ is a square diagonal matrix, for $H=L'W'WL$ the inverse equals to $H^{-1}=L^{-1}W^{-1}{W'}^{-1}{L'}^{-1}$. The question is that is there any ...
3
votes
1answer
28 views

Least squares / residual sum of squares in closed form

In finding the Residual Sum of Squares (RSS) We have: \begin{equation} \hat{Y} = X^T\hat{\beta} \end{equation} where the parameter $\hat{\beta}$ will be used in estimating the output value of input ...
1
vote
1answer
17 views

diferences of spectral decomposition of different types of matrices

For an $n \times n$ square complex matrix let say $A$ with eigenvalues $\lambda_1,\lambda_2,.....,\lambda_n$. $A$ is normal iff $A$ is unitary diagonalizable;that is there exist unitary matrix U such ...
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0answers
7 views

Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
5
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1answer
43 views

Matrix Help: Combinations

Given a 10 by 10 matrix filled with 0s and 1s, how many possible outcomes are there? It sounds easy enough as a combination of $2^{100}$. The kicker to the question is there MUST be exactly five 1's ...
0
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1answer
31 views

Matrix representation of complex numbers in exponential form

Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ...
2
votes
2answers
32 views

Uniqueness of Singular Values

Given a matrix A, one inductively constructs (and thus proving its very existence) the singular value decomposition as follows: take $ \sigma_{1}=||A||_{2} $, and consider a couple of vectors such ...
0
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0answers
25 views

Geometrical interpretation of the following transformation

I am learning linear algebra from Linear Algebra by Hadely and I came across this question that I do not have any idea how to solve Interpret geometrically the transformation produced on $E^2$ by ...
10
votes
1answer
62 views

Largest determinant of a real $3\times 3$-matrix

What is the largest determinant of a real $3\times 3$-matrix with entries from the interval $[-1,1]$ ? A result of John Williamson says that the largest value is equal to $4$, if the entries are just ...
0
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0answers
12 views

Rigid Deformation

I'm trying to parse through this paper on using the method of moving least squares for rigid transformations - http://www.cs.rice.edu/~jwarren/research/mls.pdf Under section 2.3, the author mentions ...
0
votes
1answer
25 views

Construct matrix of ones and zeros based on sequences

We are given ($a_1,a_2,....,a_m$) and ($b_1, b_2,....,b_n$) sequences with non-negative integers. Decide whether it's possible and if it is construct a matrix $\Re^{m x n}$ of ones("1") and ...
3
votes
1answer
35 views

if $tr(A)=0$,then we have $A=BC-CB?$

if for any matrix $A_{n\times n}$,and such $tr(A)=0$,show that there exist matrix $B$ and $C$ such $$A=BC-CB$$ I know prove this: if $A=BC-CB$,then we have $tr(A)=0$ because $$tr(BC)=tr(CB)$$ ...
0
votes
1answer
25 views

matrix derivative of 3 multiplied matrices

I have the following function $L=\mu_w^T\Sigma_w^{-1}\mu_w$ where both $\mu_w$ and $\Sigma_w$ are functions of a vector $w$. How do I differentiate this wrt. $w$. $\Sigma$ is a positive definite ...
0
votes
1answer
20 views

Simple expression for the number of elements in a diagonal half of a matrix

Consider the following matrix: $\begin{bmatrix} 0 & 0 & 0 & 0 \\ I & 0 & 0 & 0 \\ A & I & 0 & 0 \\ A & A & I & 0 \\ A & A & A & I ...
0
votes
1answer
18 views

Differentiate matrix quadratic

I wish to differentiate $x^TAx$ wrt. $x_i$ where $x_i$ is the i-th element in the vector $x$. I realise when differentiating wrt. $x$ alone the answer is $2Ax$. How would this change when its $x_i$
2
votes
1answer
48 views

Derivative of a Matrix to a Power

Fix a positive interger $k$ and let $F: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$ be the map on $n \times n$ matrices defined by $F(A)= A^k$. Show that $F$ is differentiable at ...
2
votes
1answer
46 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
0
votes
0answers
22 views

PSD matrix and non-negative polynomial

So I'm trying to prove that if there exists a $5 \times 5$ matrix $Q$ such that $$Q \succeq0,\,\, a_{l-1} = \sum\limits_{i+j=l} Q_{ij} , l=1,\ldots,5$$ then there exists a fourth degree polynomial ...
0
votes
1answer
25 views

Any transformation in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$. [closed]

$V$ is a two-dimensional vector space over a field $\mathbb F$; prove that every element in $A(V)$, the ring of linear transformations on $V$, satisfies a polynomial of degree $2$ over $\mathbb F$.
2
votes
1answer
29 views

Boolean Least Squares semidefinite relaxation

So I'm working on the Boolean least squares problem that comes up a lot in circuit design. In its raw form, it looks like this, $$\phi = \min \operatorname{trace}(A^TAX) - 2b^TAx + b^Tb$$ s.t. $$X = ...
2
votes
1answer
28 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
2
votes
3answers
31 views

How are specific linear maps defined?

I'm revising for exams and a question that often crops up is: given a linear map $\mathcal{T}:\;\mathbb{R}^n\to\mathbb{R}^m$, describe how to represent $\mathcal{T}$ as a matrix relative to bases ...
0
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0answers
19 views

What changes where made on this Gaussian-Elimination?

in the Internet I have found the following use of the Gaussian Elimination method: $z \in \mathbb{R}, \ n\in\mathbb{N}, n \ge 2$ and $\begin{pmatrix} z & 1 & \dots & 1 & 1 \\ 1 & ...
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0answers
24 views

markov chain computation

I consider a 2 state Markov chain: $X = \{1,2\}$, transitions are $M(i,j)$ and the matrix has a unique stationnary distribution $\pi$: $$ \pi(1) = \frac{M(2,1)}{2-M(1,1) - M(2,2)} \\ ...
0
votes
0answers
30 views

Solving Matrix Value and Optimal Strategy (Matrix Games)

How would I solve this matrix game ? I'd like to find the value of the matrix and the optimal strategies for each player. $$ \left[ \begin{array}{cccc} 0 & 3 & -2 & 2 \\ -3 & 0 ...
0
votes
1answer
27 views

proof of positive semi-definiteness of the precision matrix (inverse of the covariance matrix)

I would like to know how to prove that the inverse of a covariance matrix $\Sigma^{-1}=\Omega$ is positive semi-definite too. My second question can we prove that for any matrix $A\in\cal{M}_{nm}$ we ...
4
votes
1answer
35 views

What are eigenvalues of higher order finite differences matrices?

I know (at first empirically, then read somewhere) that for for second order finite differences matrices like $$\begin{pmatrix} -2&1&0&0&0\\ 1&-2&1&0&0\\ ...
1
vote
1answer
45 views

Matrices rank problem

$X\in \text{Mat}_n (\mathbb{R} )$ and $|X|\neq 0$. $X$ has column vectors $X_1,X_2,\ldots ,X_n$. $Y$ is a matrix that consists of column vectors $X_2,X_3,\ldots ,X_n,0$. Let $A=YX^{-1}$ and ...
0
votes
1answer
36 views

Determinant of Matrix is different than product of diagonal

(sorry in advance, but I can't find a page on how to format math equation/structures) I'm having a bit of an issue with this matrix and finding its determinant. I know what the correct determinant is ...
2
votes
2answers
39 views

Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...