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 views

A hard exercise on endomorphisms and determinants

The following exercise has been bugging me for some days, could someone help me with it ? Let $E$ be a $\mathbb{C}$-vector space with dimension $n$ and $f\in\mathcal{L}(E)$ ($\mathcal{L}(E)$ denotes ...
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0answers
8 views

Generator matrix of a Reed-Muller code

I need to find a generator matrix (2,4) of the Reed-Muller code (2,4), the dimension of R(2,4) and the minimum distance of R(2,4). I know that R(r,m) of order r, then length: n^m, dimension k = 1 + ...
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2answers
20 views

Positive definiteness of block matrices

I really appreciate if anyone can help me regarding my problem. I have a matrix in the format $M = \left[ {\begin{array}{*{20}{c}} {\delta I}&A\\ {{A^T}}&kA \end{array}} \right]$ where $A$ is ...
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0answers
15 views

Recurrence Derivative

I have a recurrence relation as follows $ \left\{ \begin{array}{ll} R_0=H & \mbox{if } n = 0 \\ R_1(s)=sR_0 \hspace{.1cm} A & \mbox{if }n=1\\ R_{n+2}(s)=\frac{s}{n+2}\{ ...
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1answer
17 views

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am ...
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1answer
9 views

Properties of Hermitian and Positive Definite matrix

Let $A \in \mathbb{C}^{nxn}$ be Hermitian and positive definite. I have to show that $|a_{jk}|^2 < a_{jj}a_{kk}$ $max_{i,j=1,\dots,n}|a_{ij}| = a_{kk}$ for some $k$ with $1\leq k\leq n$ For ...
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0answers
30 views

Trace of a certain matrix

Let $A$ be a $227 \times 227$ matrix having distinct eigenvalues , with entries from $\mathbb Z_{227}$ , then what is the trace of $A$ ?
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1answer
18 views

How to prove or understand this linear algebra assertion?

Given a matrix $B \in \mathbb{R}^{n \times k} $, and $B$ has rank $ k $. Therefore there exists a nonsingular matrix $A=( A_{1},A_{2}) \in \mathbb{R}^{n \times n} $ such that $$ AB= \left[ ...
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0answers
30 views

Proving $A_{n}$ is not invertible for n>2 when the entries are sequential integers

Let $A_{n}$ be the nxn matrix whose entries are the integers 1, 2, 3,..., n-1, n, written in order from left to right, top to bottom. For example, $$A_{5}=\begin{bmatrix} ...
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0answers
27 views

Partial Sum to be invertible [on hold]

Let $A_1,\cdots,A_m$ be $n\times n$ matrices, satisfying $$m>n, A_1+\cdots+A_m=E_n,$$ where $E_n$ is the $n\times n$ identity matrix. Show that there exists a subset $P\subset \{1,\cdots,m\}$ ...
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0answers
13 views

direct product of three square matrix

Suppose that $I_1$ is a $n_1\times n_1$ identity matrix and $I_2$ is a $n_2\times n_2$ identity matrix, and $H$ is $n\times n$ matrix. If $$ \bar H=I_1\otimes H \otimes I_2, $$ and we regard all the ...
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2answers
51 views

Is the Square Root of an Inverse Matrix Equal to the Inverse of the Square Root Matrix?

I know in general that if a matrix $A$ is positive definite, then there exists a (unique?) square root matrix $B$, which is also positive definite, such that $BB=A$. Therefore, suppose $A$ is ...
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0answers
34 views

Find a basis for symmetric $2 \times 2$ matrices [on hold]

Find a basis for the space of all $2 \times 2$ symmetric matrices. I do not even know how to start. please explain it to me step by step
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1answer
24 views

How do I rearrange this matrix equation to find X?

Given that the matrices $D$, $E$ and $F$ are invertible, how do I rearrange the equation to solve for $X$ when $D(X+3I)E = 5D(F+E) +E^2$. Would I just take the inverse of $D$ and $E$ to both sides ...
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0answers
13 views

Norm of product of two matrices

Let $A\in\mathrm{R}^{n\times n}$ and $B\in\mathrm{R}^{n\times n}$ be two matrices. If $\|\cdot\|$ denotes the matrix norm, are the followings true? $\|AB\| = \|BA\|$ $\|A^2\| = \|A\|^2$ If they ...
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1answer
15 views

Matrix diagonalization example

I want a real world example or simply a good example that explains the use of a diagonal matrix, and when to prefer to use a diagonal matrix? any other important information about diagonal matrix or ...
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0answers
20 views

Understanding the Cholesky decomposition

I'm attempting to understand the Cholesky decomposition via the following site: http://en.wikipedia.org/wiki/Cholesky_decomposition If I have a matrix, say $$A = \begin{bmatrix} 2 & -1 & ...
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0answers
6 views

Summation of all combination

I have two matrix. A=[1 2 3];B=[4 5 6]; the all possible combination of their summation is [1+4 1+5 1+6; 2+4 2+5 2+6;3+4 3+5 3+6]. Now instead of 1*3 my matrix dimension is 1*n. and instead of two I ...
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0answers
17 views

Wronskian of a fundamental set of solutions

Consider the system of equations: $$\dot x_1=x_2$$ $$\dot x_2=-q(t)x_1-p(t)x_2$$ (Sorry I don't know how to do subscript notation for the 1's and 2's, an edit would be appreciated. Also the $x_1$ ...
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1answer
27 views

Finding similar matricies

I'm trying to find a matrix N similar to the scalar matrix M = $ \begin{pmatrix} a & 0 \\ 0 & a \\ \end{pmatrix} $ Such that $M = ANA^{-1}$. I have no idea ...
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0answers
10 views

Markov chains: identifying a nonregular transitional matrix

I am currently TA'ing for a course in which the students are soon to learn about Markov chains and stochastic matrices. During the sections, it refers to the possible existence of a stable state and ...
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1answer
19 views

Prove that the determinant of this matrix is non-zero.

Prove that the determinant of this matrix is non-zero for every possible combination of + and - .$$\left[\begin{array}{cc} \pm 1 & \pm 3 & \pm 4 \\ \pm 3 & \pm 2 & \pm 5 \\ \pm 4 ...
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1answer
15 views

Some operation like determinant

we have determinant operation that is like below: $ det(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}) $= $ (-1)^{1+1}a(ei-fh)+ ...
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2answers
31 views

Determine the values of c for which the equation Ax = b is consistent.

Determine the values of c for which the equation Ax = b is consistent. A= ...
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1answer
19 views

Positive definiteness of n'th power of a positive definite matrix

Let's define a real (not necessarily symmetric) matrix $A$ to be positive definite iff $A + A^T$ is a symmetric positive definite matrix. Then can we conclude that $A^2$ or in general $A^n$ is ...
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1answer
13 views

Is the pair controllable/observable?

The matrices $Q\in\mathbb R^{n\times n}$ and $G\in\mathbb R^{n\times n}$ are both symmetric positive semidefinite, $A\in\mathbb R^{n\times n}$ is invertible. Moreover, $(A,G)$ is controllable, and ...
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5answers
39 views

Whether a matrix is a zero matrix

If a real square matrix $A $ is similar to a diagonal matrix and satisfies $A^n=0$ for some $n\in \mathbb N $,then can it be proved that $A$ must be a zero matrix?
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1answer
35 views

What are these tick marks after the x, y, and z called?

What are these marks called and what do they stand for? This is for a Affine Transformation.
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0answers
9 views

ODE -parabolic cylinder functions

How do we solve $\frac{d^2f}{dz^2} + \left(Az^2+Bz+C\right)f=0 \tag 1$ where $f(z),A,B,C$ are matrices of order $3 \times 3$.
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0answers
21 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
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1answer
30 views

Know eigenvalues, get $Q$ of $A=QLQ'$

$A=\begin{bmatrix} 1 & -2 & 2\\ -2 & -2 & 4\\ 2 & 4 & -2 \end{bmatrix}$ I have calculated that the eigenvalues $\lambda=2,2,-7$. When $\lambda=2$, the eigenvector is ...
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0answers
11 views

Converting from X,Y,Z offset representing a rotation to matrices

I've been working on figuring out 3-dimensional rotations for graphics and I've reached a brick wall of understanding that I can't power through. Right now I have a function which calculates the ...
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1answer
20 views

Proof of upper triangular matrices

I am supposing that $A=(a_{ij})$ and $B=(b_{ij})$ are two $n\times n $ upper triangular square matrices. $\lambda \in \mathbb{R}$. So $a_{ij}=0$ whenever $i>j$. I am trying to prove that these are ...
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1answer
12 views

What is the change of basis in 2D?

I know how to apply a change of basis in 1D, but I was wondering: If I want to apply a change of basis to a nxn matrix, is it enough to apply the change of basis to every column of the matrix or is ...
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2answers
68 views

Row reduction and the characteristic polynomial of a matrix

Can you row reduce the matrix before computing $\det(\lambda I-A)$? Will this still give an equivalent characteristic polynomial?
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2answers
43 views

Form a basis for R^3? [on hold]

This is a homework problem and I need help on. Consider the matrix with the given vectors as its columns. Do (1, -1, 3), (-1, 5, 1), (1, -3, 1) form a basis for R^3?
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0answers
15 views

Prove or disprove that the product $PVPVP$ is nonnegative

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a nonsingular symmetric M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is nonnegative. I know ...
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0answers
10 views

Determining matrix in terms of determinants of other matrices.

Determine |a+b e-f| |c+d g-h| in terms of the determinants of |a c| |b d| |a c| |b d| |e g| |e g| |h f| |h f| ...
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0answers
10 views

why is the covariance matrix of a bekk model always positive definite?

The BEKK(1,1) model is given by: $$\Sigma_{t}=A_{0}A_{0}'+A_{1}a_{t-1}a_{t-1}'A_{1}'+B_{1}\Sigma_{t-1}B_{1}'$$ where $a_{t}$ are serially uncorrelated, zero mean innovations, $A_{0}$ is a lower ...
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1answer
16 views

Coordinates of a vector under a basis in a Hilbert space?

Given an arbitrary basis $\{m_1, \dots, m_n \}$of a Hilbert space $H$ (or just think it as $\mathbb R^n$, and I think the methods should be the same) with given inner product, how can we find the ...
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1answer
33 views

Matrix raised to a power

Find $A^n$ for $n = 1,2,...$. Does $A^n$ tend to a limit? $$A= \begin{pmatrix} 4/5 & 2/5 \\ 1/5 & 3/5 \end{pmatrix}$$ I found the eigenvalues $\lambda=1,2/5$ and the eigenvectors ...
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2answers
34 views

Inversion of n x n matrix

A matrix F is given: $$ F = [e^{i\frac{2\pi kl}{n}}]_{k,l=0}^{n-1} $$ Find $$ F^{-1} $$ I know Gaussian method for inverting matrices but I suppose it doesn't apply to matrices with not given exact ...
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1answer
16 views

Diagonalizing the X and Z matrices

I've got two special matrices I'm trying to diagonalize : The Z matrix :$$\begin{bmatrix} 1&1&\cdots&1&1\ \\&&&1 \\&&\diagup \\&1 ...
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0answers
28 views

Isomorphism of vector spaces

Let $S$ be the space of all $3\times k$ matrices,$T$ be the space of all column vectors consists of seven components.If $S$ is isomorphic to a subspace of $T$ then what are possible values of $k$? I ...
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0answers
22 views

Stuck on kernel space question

I am completely stumped on what seems like a simple question. For a vector $v$ in $\mathbb R^n$, and $A$ being a $m \times n$ matrix with real entries. How do I show that $v-v$ is in $\ker(A)$? Any ...
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2answers
56 views

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true?

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true? $1.$ $A=I$ $2.$ $\det(A)=1$ $3.$ $\operatorname{trace}(A)=n$ I have counter example for the first option.For ...
0
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1answer
16 views

Invertibe matrix is a transition matrix?

It is true that all transition matrices are invertible, but does the converse hold: All invertible matrices are transition matrices? I'm asking with regard to matrices over a field, but more general ...
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0answers
14 views

Use the Kronecker delta matrix to answer question

So I have the Kronecker delta which is denoted as $\delta_{ij}$=$I$. Let $b_1, b_2, \cdots, b_n$ be a set of $n$ real numbers, I must show that: $\sum\limits_{i=1}^n b_i \delta_{ij} = b_j$ and ...
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0answers
23 views

Is there any simple way of finding a matrix which commutes with a given (say, more complicated) matrix?

Suppose I want to find the eigenvectors and eigenvalues of a hermitian matrix $A$, but $A$ is big and ugly. Is there an easy way to find another, nicer, hermitian matrix $B$, such that $AB=BA$ and so ...
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0answers
13 views

ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...