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), ...

learn more… | top users | synonyms (2)

0
votes
0answers
6 views

Build a rotation matrix that rotates 30 degrees along the axis (1,1,1)?

Why does the following image equal what it equals? Why does x,y,z equal that? 1/sqrt(3),1/sqrt(3),1/sqrt(3)
0
votes
0answers
17 views

Which curve we will get under $\mathcal{A} \in M_2(\Bbb{R})$ from a unit circle

If I have a circle: $x^2+y^2=1$, It's parametric equation is : $$\begin{cases} x = \cos\theta \\ y = \sin\theta \end{cases}$$ under some transform: $A=\begin{pmatrix} a & b\\ c & d ...
1
vote
2answers
50 views

determine signature of matrix

what is the signature of this matrix: $\begin{pmatrix} -3&0&-1 \\0&-3&0 \\ -1&0&-1 \end{pmatrix}$ ? I tried calculating them without eigenvalues; this should be done via ...
0
votes
1answer
17 views

what do these odds ratios represent?

I am reading this article in which is given the matrix of the joint probabilities of two random variables, X=$(x_1,x_2)$ and Y=$(y_1,y_2)$. Let's say they are $(p_{1,1},p_{1,2},p_{2,1},p_{2,2})$. ...
1
vote
0answers
18 views

Matrix Decomposition: Difference between Cholesky Decomposition, Eigendecomposition and Jordan Normal Form Decomposition

I recently created a related topic about the square root matrix, in case you'd like to refer to that one. Here's what we want: Consider the matrix $\Omega=E(\mathbf{u}^{\top}\mathbf{u})$, where ...
0
votes
0answers
24 views

Prove that there is a subset with an invertible sum

The question is as follows: For some $m>n$, let $A_1,\cdots,A_m$ be $n\times n$ matrices, satisfying $$ A_1+\cdots+A_m=I_n $$ where $I_n$ is the $n\times n$ identity matrix. Show that ...
1
vote
1answer
20 views

Multivariate Calculus - Partial Derivatives - Implicit Differentiation - Chain Rule

Let $z = z(x,y)$ be defined implicitly by $F(x, y, z(x,y)) = 0$, where $F$ is a given function of three variables. Prove that if $z(x,y)$ and $F$ are differentiable, then $$\frac{dz}{dx} = - ...
3
votes
0answers
24 views

Comparing/contrasting hyperbolic and Euclidean geometry - or, on how ${\rm PSO}_2(\Bbb R)$ sits inside ${\rm PSL}_2(\Bbb R)$

I am studying hyperbolic geometry, in particular comparing and contrasting it with familiar Euclidean geometry. Let $\Bbb E$ be the Euclidean plane, and $G={\rm Iso}^+(\Bbb E)$ be the group of ...
0
votes
0answers
13 views

Finding equation of plane in 3D

I was given 3 points on a plane: (5, 4, −8),(1, 6, −3) and (7, −2, 5) I was trying to find the equation of the plane and did the following: I chose two vectors to cross multiply to find the normal ...
1
vote
1answer
17 views

If $A_{n\times n}$ and $B_{n\times n}$ are both nonsingular real matrices, where $n$ is odd, show that $AB + BA \neq0$.

I have been puzzling over this for a while now. I tried to find something in the properties of nonsingular matrices as well as the properties of determinants that might relate, but so far I've found ...
1
vote
0answers
22 views

Linear systems, eigenvectors

For each of the following linear systems of differential equations, (i) find the general real solution (ii) show that the solutions are linearly independent (iii) draw the phase portrait a. $$\dot ...
0
votes
1answer
12 views

Finding hermitian conjugate and inverse of a complex matrix

I have the following matrix: $$ F = [e^{i\frac{2\pi kl}{n}}]^{n-1}_{k,l=0} \in \mathbb{C}^{n,n} $$ for $n = 1,2,3,...,i$ I need to find $F^HF$ and $F^{-1}$ where $F^H$ is a hermitian conjugate ...
-1
votes
0answers
9 views

Copositive matrices. [on hold]

Copositive matrices. A matrix X^2∈Sn is called copositive if zTXz≥0 for all z≥0. Verify that the set of copositive matrices is a proper cone. Find its dual cone
1
vote
0answers
28 views

Solving matrix equation of the form $(AX)^2+(BY)^2=D$

Is there any method that can solve the matrix equation of the form $(AX)^2+(BY)^2=D$? $A$ and $B$ are matrices, $X$, $Y$ and $D$ are column vectors. (Solve for $X$ and $Y$) I originally have two ...
0
votes
1answer
28 views

About kernel space

Both the square and symmetric matrices $A$ and $B$ are positive semidefinite. Moreover, $A-B$ is positive semidefinite and $\text{rank}(A)=\text{rank}(B)$. Based on these conditions, can we have ...
1
vote
1answer
48 views

Given the matrix $A^k$, how to get $A^{k+1}$?

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

Finding a linear transformation with respect to different bases

Let $f: \Bbb R^2 \rightarrow \Bbb R^2$ 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 ...
1
vote
0answers
21 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
38 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
22 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
votes
0answers
9 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 ...
0
votes
0answers
11 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
votes
2answers
14 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
315 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
votes
2answers
36 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
votes
1answer
35 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 ...
0
votes
0answers
13 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
30 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
vote
0answers
23 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
25 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
13 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 ...
0
votes
0answers
43 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
votes
2answers
25 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} ...
2
votes
1answer
66 views

Partial Sum to be invertible

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
16 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
vote
2answers
69 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 ...
-4
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
28 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
19 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
22 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
22 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
30 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
20 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
38 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 ...