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
19 views

If $A^k = \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & \cos kx\end{array}\right)$, how to get $A^{k+1}=$?

$$A^k = \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & \cos kx\end{array}\right)$$ $$A^{k+1} \overbrace{=}^? \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & \cos ...
0
votes
0answers
15 views

Finding a linear transformation with respect to different bases

Let $f: R^2 \rightarrow R^2$ be the linear transformation which rotates objects in the plane around the origin by 30 degrees counterclockwise. Find a matrix F for $f$ with respect to the standard ...
1
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0answers
20 views

How to find the basis of a matrix by using Gauss-Elliminaton?

I confuse that, This is my calculating process, Where i do the mistake in this process? I hope to understand this error.
0
votes
3answers
36 views

Linear system $AX=0$ has a nonzero solution

Which parameters $a,b,c,d$ satisfy the matrix $$A=\begin{bmatrix}-1 & 1&1&1 \\1 & -1&1&1\\1 &1 &-1 &1\\a&b&c&d \end{bmatrix}$$ sucht that linear system ...
0
votes
2answers
21 views

Find $x$ for which the rank is as minimal/maximal as possible

Find an $x$ in $\Bbb R$ for which rank of the matrix $$A=\begin{bmatrix}1 & 1&1&1 \\1 & -1&-1&1\\1 &-3 &-3 &x \end{bmatrix}$$ is as minimal/maximal as possible. I ...
0
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0answers
7 views

Clarifications regarding matrix transformations.

I have an equation which looks like this: Pos1 * L1 * X * L2 = Pos2 * R1 Where Pos1 and Pos2 are vectors. L1,X,L2 and R1 are matrices. I have to find the value for the matrix X. Please let me know ...
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0answers
8 views

Matrix problem in Mixed Regression

the background is $y= X \beta +e$ y=n*1 X=n*p $\beta=p*1$ e=n*1 take singular value decomposition of X $X=P \Delta Q$ $\beta=QKP'y$ K is a digaonal matrix and depending on its form can represent ...
0
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0answers
11 views

What is the computational cost of reduced row echelon and finding the null space?

I'm taking computational linear algebra, and haven't been able to find too much information about the computational cost (in terms of m=rows and n=cols) of these two routines: Reduced Row Echelon ...
5
votes
1answer
246 views

Is this group finite?

Let $G$ be a sub-group of the invertible real matrices of size $n$ (usually noted $GL_n(\mathbb{R})$), such that $\forall M\in G,M^2=I_n$ Is $G$ finite ?
0
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2answers
31 views

Diagonalise without finding eigenvalues

I am asked to find the Jordan normal form (in this case, diagonalise) the $n\times n$ matrix $M$ defined: $$M_{ij}=1+\delta_{ij}\,x$$ I am then asked to deduce the minimal polynomial, eigenvalues and ...
2
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1answer
29 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
12 views

Generator matrix of a Reed-Muller code [duplicate]

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 + ...
1
vote
2answers
25 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 ...
1
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0answers
21 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}\{ ...
1
vote
1answer
21 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 ...
0
votes
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
36 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 $A$ ?
0
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2answers
23 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[ ...
0
votes
0answers
31 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} ...
1
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0answers
30 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 $P\subset \{1,\cdots,m\}$ ...
0
votes
0answers
14 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 ...
1
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2answers
53 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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votes
0answers
36 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
0
votes
1answer
27 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 & ...
0
votes
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 ...
0
votes
0answers
18 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 $x_1$ ...
1
vote
1answer
28 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 ...
0
votes
0answers
11 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 ...
0
votes
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 ...
1
vote
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)+ ...
0
votes
2answers
35 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= ...
-1
votes
1answer
20 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 ...
0
votes
1answer
16 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 ...
0
votes
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?
1
vote
1answer
36 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.
1
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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 $3 \times 3$.
0
votes
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 ...
0
votes
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 ...
0
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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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
14 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 ...
5
votes
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?
-2
votes
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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vote
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 ...
0
votes
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 ...
0
votes
1answer
17 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 ...