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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21 views

Norm of matrix exponential

If $$\phi(t,0) = \exp(At)$$ and $$\|\phi\|<\exp(a+bt),$$ how to find the values of $a$ and $b$ (using equations)?
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1answer
10 views

Constructing a complete affine 3D transformation matrix with homogeneous coordinates.

I have been able to scale, rotate, and translate a 2D point represented by a 3x1 matrix as such: $$P = \left( \array{ x \\ y \\1 } \right)$$ The affine transformation that I apply to $P$ is this ...
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0answers
16 views

Basic Question Linear Transformation and Matrix computations

Can someone show me how to do this question? http://imgur.com/cIciHnY I'm studying for a test and this was a question off a past test. I would love to show my thoughts but I do not know how to format ...
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0answers
17 views

Linear algebra.Proof proportinal between minors and cofactors

$B$ is square matrix. Order of matrix $B$ is $n$. First $m$ lines form the matrix $C$, $rank (C)=m$.Last $n-m$ lines form fundamental system solutions of homogeneous linear equation with matrix $C$ ...
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0answers
12 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
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1answer
34 views

Linear maps and matrix coefficients

I am currently working through this page in my script: Can somebody explain what this means and how it works in practice? Perhaps if I saw an example I could follow it. Thanks for your help!
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1answer
16 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
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2answers
20 views

Looking for notation of set of all entries of some matrix?

I'm busy writing my thesis, and I'm looking for some concise notation to denote the supremum of the matrix entries of, say $A \in M_n(\mathbb{R})$. How should I do this? Looking for something like ...
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1answer
23 views

On the eigenvalues / properties of a specific matrix.

I'm not sure how to better phrase the title of the question, because I don't know the specific name of the matrix I am after, but I want to consider matrices of the form $$ \begin{align*} ...
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0answers
15 views

prove that F is dense in C(X×Y,R)?

Let X,Y be compact metric spaces. Let F= {∑ Ai fi(x) gi(y),fi∈∁(X,R),gi∈∁(Y,R), i from 1 to n }. prove that F is dense in C(X×Y,R) ? please i cant figure it out any help i will be thankful !
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25 views

Find Determinant of linear transformation

The question is Find the determinant of linear transformation Let V be the vector space of polynomials of degree at most over R, and define T:V to V by T(p(x))=p(1+x)-p'(1-x) for all p(x) in V. I ...
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1answer
22 views

Question regarding change of basis

Given two basis {$\textbf{e}_a$} and {$\textbf{e}_{a'}$}, we can have $$\textbf{e}_a = R^{b'}_a\textbf{e}_{b'}$$ $$\textbf{e}_{a'} = R^{b}_{a'}\textbf{e}_{b}$$ Substituting the second equation into ...
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1answer
30 views

Question about Symmetric matrix

Ok my book says this matrix $A = \left ( \array{ -2 & 1 \\ 1 & -3 } \right )$is symmetric. But, I don't understand b/c if it were a symmetric matrix, wouldn't it be ...
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0answers
21 views

Check if a vector b is orthogonal to column space of A

Using built-in matlab functions, how would you check if a vector b is orthogonal to the column space of matrix A given that the dimensions of A and b are correct and given that b is not in the column ...
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0answers
10 views

generalisation of Knonecker matrix product

In the Kronecker matrix product $C = A\otimes B$ we have that $C(i,j)=A(i,j)*B$ where the elements $A(i,j)$ are just numeric scalar values. What if the $A(i,j)$ are matrix operators which act on ...
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1answer
55 views

Why is the Det(a)=0 not a subspace? [on hold]

I'm reading my linear algebra textbook, and it says word for word: The following sets is not a subspace when the set of all 2x2 matrices B such that det(B)=0. I just need help trying to understand ...
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1answer
22 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. ...
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1answer
25 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} = ...
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1answer
25 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
6 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 ...
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2answers
52 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 ...
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2answers
28 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
23 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
48 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
47 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
votes
2answers
55 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 ...
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3answers
62 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
16 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 ...
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0answers
22 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
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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
29 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
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1answer
52 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 ...
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0answers
14 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) ...
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0answers
15 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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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
18 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
14 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
6 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
votes
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
28 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
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2answers
29 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 ...
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0answers
24 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
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1answer
61 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 ...
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0answers
11 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
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1answer
21 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
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1answer
34 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
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1answer
23 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
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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 ...