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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4
votes
1answer
72 views

Calculating the determinant of a matrix using its rank

Let A, B, C and D be real n×n matrices. If $$\operatorname{rank} \begin{bmatrix} \ A & B \\[0.3em] \ C & D \\[0.3em] \end{bmatrix} = n$$ then show that $$\det ...
2
votes
2answers
66 views

If $A$ is a matrix of size $n\times n$, and $A^2+A+2I=0$

If $A$ is a matrix of size $n\times n$, and $A^2+A+2I=0$, check whether $A$ is singular or not and find its inverse if it exists. I can find the inverse by simply multiplying the given equation with ...
0
votes
0answers
25 views

Simple excercise on linear transformations - confused

A Linear tranformation L in $\mathbb R^3$ with matrix $$ L_b^b = \left(\begin{matrix} 1 & 0 & 5 \\ 0 & -2 & 2 \\ 1 & -2 & 7 \end{matrix}\right)$$ and basis $b = \{ (1,0,2), ...
0
votes
0answers
16 views

Jacobian matrix of summation function

So let's say I have a function like this $(\mu_{ij})_{i,j=1,...,t;i+j>t}\longmapsto \sum_{i,j;i+j>t} \mu_{ij}$ and I need to find the Jacobian matrix of that function. I tried to calculate it ...
2
votes
1answer
40 views

What does a homomorphism $\phi: M_k \to M_n$ look like?

Let $\phi : M_k(\Bbb{C}) \to M_n(\Bbb{C})$ be a homomorphism of $C^*$-algebras. We know that $\phi$ decomposes as a direct sum of irreducible representations, each of them equivalent to the identity ...
1
vote
1answer
41 views

Diagonalization of a Matrix Function

Is it possible to diagonalize a fundamental matrix $\phi(t)$ of a differential equation $dx(t)=A(t)x(t)dt$, i.e., is it possible to use a diagonal matrix in place of the fundamental matrix?
1
vote
0answers
42 views

Grid traversal algorithm

I am trying to solve a programming puzzle that goes as follows: Imagine we have a market or mall with several stores in it. The market is represented as an NxN grid where 1 <= N <= 20 and every ...
-1
votes
0answers
28 views

Eigenvalues matrices

Let matrix $$A=\left(\begin{array}{ccc} 1 & 1 & 1\\1 & \omega^2 & \omega \\ 1 & \omega & \omega^2 \end{array}\right)$$ where $\omega$ is the cube root of unity. If $a,b,c$ are ...
3
votes
1answer
19 views

“Reverse” of frobenius matrix norm inequality

Suppose that we have some $m \times n$ matrix $C$, and its full rank (skeleton) decomposition $$ C = AB^T, $$ where $A$ is $m\times r$ and $B$ is $n\times r$ for some $r$. We know that frobenius norm ...
2
votes
1answer
21 views

Orthogonal matrix representation

If $\mathbf{M}$ is anti-symmetric, then $\mathbf{U}=(\mathbf{I}-\mathbf{M})(\mathbf{I}+\mathbf{M})^{-1}$ is orthogonal with $\det\mathbf{U}=1$. This is just manipulation and noticing that ...
0
votes
1answer
27 views

Condition number for each variable

Condition number of a matrix tells us how viable it is to solve $Ax=b$ $$A= \begin{bmatrix} 1.001&1\\ 1&1 \end{bmatrix} $$ Is a matrix that would be difficult to solve numerically. However ...
0
votes
3answers
65 views

How do I calculate generalized eigenvectors?

I have the matrix $$A=\begin{pmatrix} 5 & 1 & 0\\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix}$$ and I should determine generalised eigenvectors, if they exist. I found one ...
1
vote
2answers
40 views

Problem with finding the kernel of a linear map

Let $T$ be a linear map which is represented by the following matrix in the standard basis. $$\begin{pmatrix}-1 & 0 & 1 \\ 1 & 2 & 3 \\ 2 & 3 & 4 \end{pmatrix}$$ I'm trying ...
5
votes
2answers
58 views

Show that $\left\| \exp(A)-\mathbf{1} \right\| \leq e^{\left\|A\right\|}-1$

Have been attempting this question, just wondering if my answer looks alright. Question: Given $A \in \Bbb{K}^{n\times n}$ show that $\left\| \exp(A)-\mathbf{1} \right\| \leq e^{\left\|A\right\|}-1$ ...
1
vote
1answer
48 views

Adjoint of an adjoint of a matrix

Can you please help me on this question? $\DeclareMathOperator{\adj}{adj}$ $A$ is a real $n \times n$ matrix; show that: $\adj(\adj(A)) = (\det A)^{n-2}A$ I don't know which of the expressions ...
1
vote
1answer
27 views

What is the name of the 3D matrices?

The name of a variable in the $\mathbb{R}$ is called scalar. Multiple scalars form a vector: $\mathbb{R}^n$ Two or more vectors form together a matrix: $\mathbb{R}^{n \times m }$ But what is the ...
1
vote
1answer
31 views

Numerical Range of A and A transpose.

I was playing around with the numerical range [NR] (or field of value) of a matrix $A \in \mathbb{C}^{n\times n}$ lately. And was actually looking for a proof to show: \begin{equation} A=A^H : F(A) = ...
1
vote
0answers
12 views

In which cases are the main diagonal elements of a product of positive definite matrices positive?

Let $A$ and $B$ be symmetric positive definite (pd) $n \times n$ matrices and $C = A \cdot B$. In which cases is then every $c_{ii}$, the $i$-th main diagonal elements of $C$, positive? When $A$ ...
0
votes
0answers
30 views

Does is matter whether you write a matrix using vector columns or rows?

given say a set of vectors in R^3. (1,1,2) , (-1,-1,3) , (6,2,2) Does is make any difference if the vectors are written as rows to form a matrix or columns? I just ask because one lecturer will write ...
2
votes
0answers
32 views

Evaluate the product $\DeclareMathOperator{tr}{tr}\tr(AB)\tr(CB^{-1})$

Let $A,C$ be given positive semidefinite matrices, $B$ be an arbitrary positive definite matrix. How can I estimate the value of $\tr(AB)\tr(CB^{-1})$ ? Is that true $\tr(AB)\tr(CB^{-1}) \geq \tr(AC)$ ...
0
votes
0answers
16 views

Optimize matrix multiplication when one matrix is the same.

I have a situation where I will be multiplying $AB\vec{x}$ together frequently. $A$ is a 4x4 matrix that won't change from problem to problem. $B$ is a 4x4 matrix that will change occasionally, and ...
6
votes
1answer
132 views

Eigenvalue of block matrix of order $2n$

How to find eigenvalues of following block matrix? $$P=\begin{bmatrix} A & B \\ B & A \end{bmatrix}$$ Where, $A=\begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & \cdots & ...
0
votes
0answers
29 views

Associated matrix with the linear map of identity mapping [closed]

The problem is an example from the book "Linear Algebra" by Serge Lang and I think that the answer might be wrong. The question is: Let $\textit{id}:V \rightarrow V$ be the identity map. Then for any ...
0
votes
0answers
24 views

Showing Left Side to Right Side.

Let $\mathbf x$ is a $(p\times 1)$ vector, $\mathbf\mu_1$ is a $(p\times 1)$ vector, $\mathbf\mu_2$ is a $(p\times 1)$ vector, and $\Sigma$ is a $(p\times p)$ matrix. Now I have to show ...
3
votes
3answers
51 views

How do I find the Jordan normal form when I only have one eigenvalue?

I have matrix $A=\begin{pmatrix}3 &1 \\ -1 &1 \end{pmatrix}$. I have found that the eigenvalue is $2$ and the eigenvector is $\begin{pmatrix}1\\ -1\end{pmatrix}$. How do I find $T$ so that I ...
3
votes
1answer
52 views

What is the relationship between the rank of $C_i$ and the rank of $A,B$?

Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ The ...
0
votes
0answers
16 views

Bounding off-diagonal entries of a matrix

The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $Y = \begin{bmatrix} 0 & -i \\ i & 0 ...
0
votes
0answers
36 views

Proof that $\sum_{j=0}^\infty C_j$ converges if $\sum_{j=0}^\infty \|C_j\|$ converges

$C_j$ is a sequence of matrices in $\mathbb C^{n \times n}$ and the identity $$\max_{j,k}|A_{j,k}|\leq \|A\|\leq n\max_{j,k}|A_{j,k}|$$ is known. Show that $\sum_{j=0}^\infty C_j$ converges if ...
0
votes
1answer
25 views

Proving Transformation defined as a matrix is linear

I came across a problem thatwhile doing some review that states: Consider the transformation $\textit{T}$:$\mathbb{R}^2\rightarrow\mathbb{R}^2$ defined by the matrix $$ \begin{pmatrix} ...
11
votes
4answers
733 views

Is there an operation on matrices such that the determinant yields a homomorphism with the additive group of the reals?

It well known that, under standard matrix multiplication $\det(AB) = \det(A)\det(B)$, or in other words, that $\det : \mathbb{R}^{n \times n} \rightarrow \langle\mathbb{R}, * \rangle$ is a monoid ...
0
votes
1answer
11 views

2-order curve main axis transformation

I have to do main axis transformation this curve: $3x^{2} -8xy-3y^{2}=c $ I did transformation and got this: $\frac{5x^{2}}{c}-\frac{5y^{2}}{c}=1$ I think it should be correct. Now I have to find ...
1
vote
1answer
34 views

Are covariance matrices guaranteed to have real-valued eigenvectors and eigenvalues (related to PCA)?

I'm a software engineering, and I'm trying to learn principal components analysis. I've read that as part of PCA, I need to compute a covariance matrix and then find the eigenvectors and eigenvalues ...
0
votes
0answers
20 views

Quadratic polynomials in variables x,yx,y

I have two equations which define 2-order areas I have to determine matrix, eigenvalues, definiteness and area. I determine first three things, but how can I find out which area is it? Example 1. ...
0
votes
0answers
9 views

SVD matrix properties after blockwise scalar multiplication

Are there any findings relating the singular value decomposition (SVD) of a block matrix to the SVD of the same matrix after multiplying each block with a constant ? In other words, let $A \in ...
0
votes
2answers
25 views

How this non-zero matrix can be transformed to a row-echelon matrix?

The procedure introduced in the proof of the following theorem, probably does not work generally! Given the matrix $$A= \left( \begin{array}{ccc} 1 & 1 & 0 & 1 & 4 \\ 2 & 0 ...
1
vote
0answers
22 views

Matrix vector product $O(n)$

Consider the matrix vector product $x = Lb$ where $L$ is an $n\times n$ unit lower triangular matrix with all of its nonzero elements equal to $1$. For example if $n = 4$ then \begin{align*} x ...
0
votes
1answer
42 views

Algebra of matrices [closed]

How to find the possible square roots of the two rowed unit matrix I ? I took a matrix like this $$A=\begin{pmatrix} a& b \\ c & -a \end{pmatrix}$$ I couldn't figure out why $b$ and $c$ ...
4
votes
2answers
72 views

Matrix equivalent to linear maps - sanity check

I'm reading some Linear algebra notes I found online, and am a bit confused about the following: If $U,V$ are finite dimensional $\mathbb{C}$-spaces with bases $(\mathbf{u}_1,\dots,\mathbf{u}_m)$ and ...
0
votes
1answer
22 views

Network energy function matrix representation

The question seems very simple, however I`m trying to find the right blas-function which correctly describes the following expression: $\sum_{i=1}^{N}\sum_{j=1}^{N}{w_{ij}x_ix_j}$ Is it ...
1
vote
1answer
42 views

Finding the eigenvectors (& describing the eigenspace) of a Householder transformation matrix

If one is asked to find the eigenvector(s) for a Householder transformation matrix, but one is not given the values of or dimensions of the unit vector $u$. So if $H = I_n - 2uu^T$ where $I_n$ is the ...
1
vote
1answer
25 views

One parameters group generated by a real matrix

For a matrix $A\in M_n (\mathbb R),$ we consider the exponential $e^{tA}, t\in \mathbb R$. For $x\in \mathbb R^n\setminus\{0\},$ let $f : t\longmapsto e^{tA}x.$ My question concerns the surjectivity ...
2
votes
0answers
39 views

Generators of a matrix group

I have just read that the group $\Gamma_1(6)=\left\{ \begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z}):a\equiv d\equiv 1 \mod 6,\, c\equiv 0\mod 6\right\}$ is generated by the matrices ...
0
votes
0answers
10 views

How to translate between two 3D coordinate systems

I am working with 3 coordinate systems: Blender, Bullet Physics engine and LibGDX. I use Blender as UI engine with it's built in Bullet Physics and exporting those to LibGDX. I am not sure, but I ...
1
vote
2answers
226 views

Prove that an $n \times n$ matrix $A$ over $\mathbb{Z}_2$ is diagonalisable and invertible if and only if $A=I_n$

Through some facts, when $A$ is invertible, I found out that the eigenvalue can't be $0$, since if the eigenvalue is $0$, then $\det(A)=0$, which means that is is not invertible. Since it is over ...
0
votes
0answers
39 views

Two-dimensional cyclic matrices as arguments for functions

I am looking for a proof or a counterexample. Let $(a;b):=\pmatrix{a&b\\b&a}$. $k\in\mathbb{N}_0$. $$ \Longrightarrow (a;b)^k= ...
0
votes
1answer
18 views

Lagrange method with vectors?

How does one apply the Lagrange method to vectors? The problem I have (it's financial) is $$\max_w w^T r - w^T\Sigma ^{-1} w $$ under the condition that $w_1 + w_2 = 1$. $r$ is a known vector with 2 ...
0
votes
1answer
25 views

Lineaire transformation of matrices, how to tackle

I've been learning linear algebra but can't understand the concepts of linear transformation. Correct me where i'm wrong: Say i'm given $T:R^2 \longrightarrow R^2$ is my transformation. This tells ...
1
vote
1answer
26 views

rank $r$ lambda matrix and relationship of its minors of order $\le r$

Could anyone help me to prove the following or at least make me understand with an example? $A(\lambda)$ be any $\lambda$ matrix (A matrix whose entries are polynomial in variable $\lambda$) of rank ...
1
vote
1answer
37 views

Infinite matrix product

Let $$X=\left(\begin{array}{c} x_1 \\ x_2\\ \vdots \end{array}\right)$$ be an infinite real vector and $$A=(a_{ij}), \ 0<i,j<\infty$$ be an infinite real matrix. (1) For which $A$ can one ...
0
votes
1answer
57 views

Product of the differences of two pd matrices and their respective inverses is pd

Given two $\textbf{positive definite (pd), Hermitian}$ matrices X and Y, I am trying to determine whether $(X-Y)(Y^{-1}-X^{-1})$ will always be pd as well, and how to prove this. This formulation ...