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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2
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3answers
21 views

Inverse Matrices and Infinite Series

Given that $C=I+A+A^2+A^3+ \ldots$ Prove that I-A is the inverse of $C$ Hint: Use the infinite series technique for finding inverse of a matrix. Now I know with an infinite geometric series with a ...
1
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1answer
14 views

How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix/Vector Expression

Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that \begin{align} \sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i &= \begin{bmatrix} ...
0
votes
0answers
19 views

Does this guassian elimination have a solution?

I was asked to find the following solutions using guassian elimination, but I was unsure of my answers since it became quite messy but the variables still somehow fit: $$\left[\begin{array}{ccc|c} ...
-2
votes
0answers
17 views
0
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1answer
21 views

Prove True or false : If A and B are nxn invertible matrices and (AB)^2=A^2B^2, then AB=BA

This looks like it is false but the thing is I can't find a counter example for it.
1
vote
1answer
17 views

Number of positive, negative eigenvalues and the number of sign changes in the determinants of the upper left submatrices of a symmetric matrix.

How do we prove that the number of sign changes in the sequence of the determinants of the upper-left matrices of a symmetric matrix $A$ corresponds to the number of positive and negative eigenvalues ...
1
vote
1answer
15 views

How to prove that an M-matrix is inverse-positive?

Wikipedia says that The inverse of any non-singular M-matrix is a non-negative matrix." To be more precise, if $A$ is an M-matrix, then the entries of the inverse of $A$ are all non-negative, ...
1
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2answers
18 views

Orthogonal Matrices and Similarity Transforms

Sorry I can't be more specific with the title. I really don't know what to call this and about 2 hours of Googling has yielded no results. All we are given: $U$ is $n\times n$ and orthogonal $Ax = ...
0
votes
1answer
12 views

Covariance matrix of Y when we have the covariance matrix of X

If the random vector $\mathbf{X}$ is transformed according to \begin{align*} Y_1 &= X_1\\ Y_2 &= X_1 + X_2 \end{align*} and has a covariance matrix $$ \mathbf{C}_X = ...
3
votes
1answer
14 views

For what kind of matrix does it hold $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
-2
votes
0answers
25 views

Prove that theorems about trace of matrix: [on hold]

$ 1-) $    If A is an nxn symmetric matrix with r nonzero characteristic roots $ \lambda_1,\lambda_2,...,\lambda_r $, then $ tr(A^-)=\sum_{i=1}^r \lambda_i^{-1}$ $ 2-) ...
-3
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1answer
26 views

Prove that theorem about trace of non-negative matrix: [on hold]

If A is a non-negative nxn matrix , then $\ tr(A) = 0 $ if and only if $ A =0$
1
vote
2answers
25 views

How to show that $\| QA\|_2=\| A \|_2$ where $Q$ is unitary (for a matrix A)

I want to show that for a unitary matrix $Q$ and a matrix $A$ that $$ \|QA\|_2=\|A\|_2$$ I start with the definition of matrix induced norms: $$\| QA \|_2 = \sup_{x \neq ...
0
votes
1answer
11 views

Are the following vectors in the range of A

Let $A$ be the following matrix: $$ \left( \begin{array}{cccccc} 1 & 2 & 1 & 3 & 2 & 1\\ 2 & 0 & 3 & 2 & 3 & 0 \\ 4 & 2 & 1 & 1 & 2 & 1 \\ ...
0
votes
2answers
21 views

Number of Jordan canonical form of a matrix

Let, $A\in M(3,C)$. Assume that the characteristic & minimal polynomial of $A$ are known. Then what is the number of possible Jordan form of $A$ and how? What changes if we replace $C$ by $R$ or ...
5
votes
2answers
30 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
0
votes
2answers
18 views

'A' transpose inverse equals to 'B' transpose

I searched everywhere but I could not find a solution to this problem. Let $A$ and $B$ be invertible matrices with $AB = I$. Show that
0
votes
1answer
21 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
20 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
51 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
20 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
26 views

Matrix Decompositions: 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
39 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
21 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
31 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
23 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
13 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
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0answers
32 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
49 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
25 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
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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
40 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
23 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
12 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
18 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 ...
6
votes
1answer
319 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
38 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
42 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
33 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
26 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 ...
1
vote
1answer
17 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 ...
-1
votes
0answers
47 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
27 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[ ...