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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0
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2answers
90 views

Proof: $Ax=x$ for all $x$ implies $A=I$ [on hold]

Let $A$ be a square matrix of order $n$ and let $x$ be an $n$-vector. Prove that if $Ax=x$ for all $x$, then $A=I$. Thanks in advance
1
vote
0answers
24 views

How to find the inverse of this particular symmetric matrix

Basically, I have a $n \times n$ symmetric matrix, which looks like this: $$ \begin{bmatrix} 1 & \alpha & \cdots & \alpha \\ \alpha & 1 & \cdots &\alpha \\ \vdots &\vdots ...
1
vote
0answers
19 views
1
vote
1answer
203 views

Schur complement condition for positive definiteness still for complex matrices

Say that we're given the following matrices: $S\in M_{m,m}$ symmetric positive definite, $A\in M_{n,n}$, $X\in M_{n,m}$ and $Y\in M_{n,m}$. Actually I need to use a program that doesn't accept matrix ...
0
votes
0answers
23 views

Markov Transition Matrix

I have some data, shown below. How do I construct a transition matrix, for Markov Chain ? I need the formula to calculate observation data into transition matrix. Thanks! Accumulative ...
1
vote
4answers
66 views

Prove the matrix satisfies the equation $A^2 -4A-5I=0$ [on hold]

How to prove that $$ A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} $$ satisfies the equation $A^2 -4A-5I=0$?
2
votes
2answers
39 views

solving $X^2 - 3X - A = 0$ where $A,X \in \mathbb{M_2(\mathbb{R})}$

Given $A = \begin{pmatrix} 7 & 3 \\ 3 & 7 \end{pmatrix}$ find a $2\times 2$ matrix $X$ s.t. $X^2 - 3X - A = 0$, in the previous parts I have diagonalised $A$ and got $P^{-1}AP = \begin ...
13
votes
3answers
307 views

when does $\det(AB^T+BA^T)\le \det(AA^T+BB^T)$ hold?

When does the following matrix inequality hold? $$\det(AB+B^TA^T)\le \det(AA^T+BB^T)$$ $A$ and $B$ are any real matrices. My reply gives a counter example. The question is under what condition ...
0
votes
0answers
13 views

Is there a quickest method for computing QR decomposition by hand?

I'm currently studying QR decomposition and I've seen that one can arrive at $A=QR$ via the Gram-Schmidt process, using Householder matrices or using Givens rotation matrices. My question is: when ...
1
vote
1answer
31 views

Differentiation of $u^{T}Su$

I want to differentiate $u^{T}Su$ wrt $u$ where $u$ is $n$ x $1$ and $S$ is $n$ x $n$matrix . So I did the following . Since $u^{T}Su$ is a number , I wrote its expression ie $$ f = ...
0
votes
0answers
30 views

What does my teacher mean by 'choosing' from a vector?

I'm revising some lecture notes from a class I missed, I'm just struggling to figure out what she means at this point. What is choosing x1=0, x2=1... etc mean? Could someone explain?
2
votes
2answers
38 views

Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
2
votes
1answer
56 views

Why is the permanent of interest for complexity theorists?

Studying a bit about the determinant and the permanent, I'm told that although both concepts have very similar formulas, the permanent was of not much interest historically - it was until later that ...
2
votes
0answers
18 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
1
vote
2answers
23 views

induction on matrices with powers + addition and limit

$A= \begin{bmatrix} 1-q && p \\ q && 1-q \end{bmatrix}, 0<p<1, 0<q<1,$ Using mathematical induction show that $A^n$ = $\frac{1}{p+q}\begin{bmatrix} q && p \\ q ...
1
vote
3answers
35 views

Standard matrix A of T?

Help please. What would be the standard matrix of A? I know how to do number 2 and 3 but I'm just having trouble with A. I asked this earlier but I lost my account and I'm not sure if I posted ...
2
votes
1answer
27 views

Givens rotation and retraction mapping

Here's part of the texts in the book Optimization Algorithms on Matrix Manifolds, when talking about the retraction on a matrix sub-manifold. A retraction is a mapping that maps a tangent vector in a ...
0
votes
0answers
10 views

Using basic transformations to derive matrix for the reflection in a line?

Using basic transformations (translation, scaling and rotations), show all the steps to derive the transformation matrix for the reflection of points n the line : y = 3 - x I know that a directional ...
1
vote
1answer
15 views

A question about matrix spectrum property

Suppose $x\in\mathbb{R}^n$, $A\in\mathbb{R}^{n\times n}$. Does anyone know the answer to the following problems. (1) $\min\limits_{x\neq0} f(x)=\frac{x^\mathrm{T}A^\mathrm{T}Ax}{x^\mathrm{T}Ax}$, ...
0
votes
1answer
41 views

Another matrix of a given operator $A \otimes A$

Let $V$ be a $4$-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $(e_{1}, ...
1
vote
1answer
62 views

Is this function involving matrices convex?

Let $X\in \mathbb{R}^{n \times n}$. Then, is the function $$ \text{Tr}\left( (X^T X )^{-1} \right)$$ convex in $X$? ($\text{Tr}$ denotes the trace operator)
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0answers
16 views

linear binary code problem

Let $\mathcal C$ be a $[n,k,d]$ linear binary code such that $\mathcal C$ has a systematic generating matrix $G=[I_k\mid A]$. (i) Prove that $u\in (\mathbb F_2)^k$ is coded by $c=(u\mid uA)\in ...
1
vote
1answer
20 views

Estimate $\frac{(\frac{1}{2}\lambda_{\text{min}}(A+A^T))^2}{\lambda_{\text{max}}A^TA}<1$

If $A$ is positive definite, (maybe not symmetric), how to prove that $$\frac{(\frac{1}{2}\lambda_{\text{min}}(A+A^T))^2}{\lambda_{\text{max}}A^TA}<1,$$ I know that ...
1
vote
3answers
22 views

How to get An eigenvalue and eigenvectors of a matrix that contain both zero column and zero row?

Could anyone help in how to get the eigenvalue and eigenvectors of a matrix that contain both zero column and zero row like : \begin{pmatrix} -1 & 1 & 0\\ 1 & -1 & 0\\ 0 ...
0
votes
3answers
30 views

For two p.d. matrices $A$ and $B$, prove that $\lambda_1(AB)\leqslant \lambda_1(A) \cdot\lambda_1(B)$

If $A$ and $B$ are two nxn positive definite matrices, then show that $$\lambda_1(AB) \leqslant \lambda_1(A) \cdot \lambda_1(B),$$ where $\lambda_1(\cdot)$ denotes the largest eigenvalue.
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votes
0answers
13 views

The lower bound of Cheeger Inequality as the degree goes to infinity

Consider an undirected graph $G(V,E)$ with adjacency matrix $A$ and define the graph Laplacian as \begin{equation} L = D - A \end{equation} where $D$ is a diagonal matrix such that $D(i,i) = d_i$. ...
4
votes
1answer
28 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 ...
0
votes
1answer
37 views

Does Fermat's Little Theorem apply to matrices?

I'm working on a problem involving applying FLT to matrices, so any information about how to do this or prove this is true would be helpful. I've been doing some research and experimenting a little, ...
1
vote
1answer
15 views

Algorithm to compute maximum permutation sum in matrix

Given a matrix $A_{n\times n}$ of real numbers, what fast algorithms do there exist to compute the maximum value of $a_{1,\sigma(1)}+a_{2,\sigma(2)}+\ldots+a_{n,\sigma(n)}$ over all permutations ...
0
votes
3answers
38 views

Find the rank of the given matrix

Let $x_1$,$x_2$,$x_3$,$x_4$,$y_1$,$y_2$,$y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a 4 x 4 matrix A by A = $$\begin{pmatrix} x_1^2 + y_1^2 & x_1x_2 + y_1y_2 ...
1
vote
2answers
73 views

The eigenvalue of $A^TA$

If $\lambda$ is the eigenvalue of matrix $A$,what is the eigenvalue of $A^TA$?I have no clue about it. Can anyone help with that?
0
votes
1answer
19 views

Find unknown matrix in equation with 3 multiplications.

A matrix $D$ is calculated as $A*B*C$. I need to find the matrix $B$ given matrices $A$, $C$ and $D$. After some trial and error it seems that the following equation is needed to reproduce matrix ...
2
votes
1answer
18 views

Determining the structure of the abelian group, integral matrix

I am revising for my upcoming university exams and I have a past exam question that I am finding particularly challenging... a) Consider the integral matrix $$R=\begin{bmatrix} 2 & 2 & ...
0
votes
1answer
11 views

Using QR decomposition to solve a system of equations with a singular matrix

If $A\in\mathbb{R}^{n\times n}$ is singular and $x,b\in\mathbb{R}^{n}$ are such that $Ax=b$, am I right in thinking that the upper triangular matrix $R$ of $A$'s $QR$ decomposition must have at least ...
0
votes
1answer
19 views

SOR method converges for $\left( \begin{array}{ccc}2& -1\\-2 & 2\end{array} \right)$

Prove that the SOR method converges in $\mathbb{R}^n$ for the matrix $\left( \begin{array}{ccc}2& -1\\-2 & 2\end{array} \right)$ iff $\omega\in(0,2)$.
0
votes
1answer
15 views

Retrieving a Matrix from a Matrix multiplication

I have made a matrix multiplication in Matlab (K, P and S are all 2x2 matrices): K = P * transpose(H)*S Now Im given K, P, and H. I need to know S. Given that ...
0
votes
1answer
2k views

Inverse of upper triangular matrix

I have an upper triangular matrix that I want to solve the inverse for. I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity ...
2
votes
1answer
20 views

Prove the matrix $\left( \begin{array}{ccc}A_{11}&A_{12}\\A_{21}&B_{22}+A_{21}A_{11}^{-1}A_{12}\end{array}\right)$ spd

Let $$A=\left( \begin{array}{ccc}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}\right)\in R^{n\times n}$$ be a symmetric positive definite matrix with blocks $A_{ij}\in\mathbb{R}^{n_i\times ...
0
votes
1answer
9 views

GMRES converges for $\left(\begin{array}{ccc}1&-1\\-10&12\end{array}\right)$ [on hold]

Prove that the generalized minimal residual method(GMRES) converges in $\mathbb{R^n}$ for the matrix $$\left(\begin{array}{ccc}1&-1\\-10&12\end{array}\right)$$ where $H=I$ and $D=$ ...
2
votes
1answer
13 views

least square approximation: how this matrix calculation equation is deducted?

I am reading a book "kernel methods for pattern analysis". For the least square approximation, it is to minimise the sum of the square of the discrepancies: $$e=y-Xw$$ Therefore it is to minimize $$ ...
0
votes
1answer
34 views

Prove that $F$ is dense in $C(X\times Y,\mathbb{R})$?

Let $X$ and $Y$ be compact metric spaces. Let $$ F= \Bigl\{\sum_{i=1}^n A_i f_i(x) g_i(y): f_i\in C(X,\mathbb{R}),g_i\in C(Y,\mathbb{R}), 1\le i\le n \Bigr\}. $$ Prove that $F$ is dense in $C(X\times ...
4
votes
1answer
252 views

Rank of matrix as a difference of ranks

If $X$ is an $n \times p$ matrix of rank $r$ and $C = AX$ for some $q \times n$ matrix $A$ with rank$(A) = q$, how do I show that rank $(X(I-C^{-}C))=$ rank$(X)-$ rank$(C)$? I can show that rank ...
0
votes
1answer
25 views

Help with Autonne-Takagi factorization of a complex symmetric matrix.

Let $A=A_1i+A_2$ with $A$ non singular. Now let $$B =\begin{bmatrix} A_1 & A_2\\ A_2 & -A1 \end{bmatrix}$$ With $A_1$, $A_2$ and $B$ symmetric. Is it true that: 1) $B$ is non singular 2) ...
0
votes
0answers
32 views

Prove that B is bounded ?? [on hold]

Let G be a Banach space and let B be a subset of G. Suppose that f∈G* we have f(B) = {f(x); x∈B} is bounded in R. Prove that B is bounded. Such that G* is the dual space.
2
votes
1answer
28 views

One step Gauss Seidel method

Apply one step of the Gauss Seidel method to $A\textbf{x} = b$ with A = $\begin{bmatrix} 4 & 2 & 1 \\ 1 & 4 & 1 \\ 1 & 2 & 4 \end{bmatrix}$, b = $\begin{bmatrix} 4\\ ...
0
votes
2answers
183 views

Is every symmetric positive semi-definite matrix a covariance of some multivariate distribution?

One can easily prove that every covariance matrix is positive semi-definite. I come across many claims that the converse is also true; that is, Every symmetric positive semi-definite matrix is a ...
2
votes
1answer
17 views

How does a cropping of a 2D matrix/image affect its DCT transform?

I apologize in advance: since I am not a mathematician, maybe my question is not well defined, but I hope that some of you will still understand my meaning. Given a 2D matrix, or an image of ...
0
votes
2answers
23 views

Uniqueness in Matrix Multiplication

I'm sure there is an answer to this somewhere else, but I'm simply not sure how to find it or what to call it. I looked online, but couldn't find anything. The question is as follows: Let $A$ and ...
0
votes
1answer
13 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 ...
0
votes
0answers
27 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)?