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
votes
1answer
13 views

Dimension of the eigenvector

Can an eigenvector have rank>1 ? I have encountered such evecs but couldnt know how to express them in S. Or I might have done some calculation mistakes.
1
vote
1answer
18 views

Find all the eigenvalues of the matrix $A$ without computing it's c. polynomial

$A=\begin{pmatrix}6&1&1&1\\ 2&7&2&2\\ 3&3&8&3\\ 4&4&4&9\end{pmatrix}$ Find all the eigenvalues of this matrix, without computing it's characteristic ...
3
votes
2answers
45 views

YX - XY = X for nilpotent matrix

Let $X$ be a matrix over $\Bbb C$, I have to show that exists a matrix $Y$ s.t. $YX - XY = X$ iff $X$ is nilpotent. What have I done? Given $Y$ exists, I have already shown that $tr(X^i)=0$ $\forall ...
0
votes
1answer
16 views

skew symmetric matrice

Let $A$ be a matrice with real entries and $A^T=-A$ with eigenvalues $\pm i\lambda$ with $\lambda\in\mathbb R_{\gt0}$ with respective eigenspaces of dimension $1$ (i.e. $\dim \operatorname{Eig}_{i\...
1
vote
3answers
43 views

In order to find $e^{AT}$

in order to find $e^{AT}$ We can't just take the exponential of A as we would do in its diagonalized form. We need to diagonalize $A=S^{-1}e^{\delta(t)}S$ in order to find $e^{AT}$ why is this the ...
1
vote
1answer
26 views

Solve the second order equation

$$\frac{d^2u}{dt^2} = \begin{bmatrix} -5 & -1 \\ -1 & -5 \end{bmatrix}u $$ with $$ u(0)=\begin{bmatrix}1 \\ 0 \end{bmatrix} $$ and $$ u'(0) = \begin{bmatrix}0 \\ 0 \end{bmatrix} $$ Okay I ...
0
votes
2answers
33 views

How to prove this inner product in matrix form

Assume $ U$ is a unitary matrix and $x\in\mathbb{C}^n$, $||x|| = 1 $ Prove: $\langle U^HTUx\ , \ x\rangle = \langle Tx\ ,\ x\rangle$ I know that $\langle U^HTUx\ , \ x\rangle = \langle TUx\ ,\ Ux\...
1
vote
1answer
35 views

How to find the eigenvectors when complex numbers come in?

$$A=\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$ When I compute the eigenvalues of A let $e$ donate the eigenvalues $$e_1=i\\ e_2=-i$$ when i put the $e_1$ in the matrix $$A-e_1I=\...
3
votes
3answers
45 views

Find the eigenvalues of a symmetric matrix

Find the eigenvalues of a $3 \times 3$ symmetric matrix with $1$ on the main diagonal and $\frac{1}{\sqrt 3}$ off the main diagonal. Since each row on addition give the same value, one of the three ...
2
votes
2answers
45 views

Showing that $I+tA$ is invertible using $f(t)=\det(I+tA)$ [on hold]

Let $A \in \mathbb{R}^{n\times n}$ a square matrix with $\text{trace}(A) \neq 0$. I would like to show that $f : \mathbb{R} \to \mathbb{R}$, $$f(t)=\det(I+tA)$$ local while $t=0$ is invertible.
0
votes
0answers
19 views

I have a problem I would like some advice with. I have to discretize bellow integration over volume in 2D.

I know what is discrete form of $$\int_\Omega\nabla\phi\,dV$$ in which $\phi$ is vector in 2D and $\Omega$ is volume of cell in CFD field. The result is: $$\frac{1}{\text{Volume}}\sum_{\text{face}} n\...
1
vote
0answers
38 views

Answer checking - involved derivative under summation

I'd like someone smarter and more experienced than me to check my answer and give advice on how to do it better and derive a closed form for what I'm looking for. Given a matrix $Y \in \mathbb R^{m \...
2
votes
0answers
37 views

A special decomposition of any positive semidefinite matrix?

Let $A\geq 0$ be an $n\times n$ non-zero PSD matrix (i.e., $x^*Ax\geq 0$ for all $x\in\mathbb{C}^n$) having rank >1. I was wondering if $A$ could always be decomposed as $$A=(1-\alpha)\,B+\alpha\, uu^*...
1
vote
1answer
4 views

Find a Matrix given a characteristic equation

How I can compute a matrix given the characteristic equation ? All I found are references and functions that do the exact opposite, but I know the characteristic equation and I need the corresponding ...
1
vote
0answers
27 views

Solving Systems of linear equations between a square matrix and a rectangular matrix with block decomposition

I am trying to decompose solving a system of linear equations using block decomposition where I have an (n x n) matrix A, which is a lower/upper triangular matrix, and a matrix B, which is a ...
0
votes
0answers
31 views

prove that $A^{-1} = (1/detA) \operatorname{cof} A^T$ [on hold]

Can you please explain to me how to prove this theorem? Theorem: if $\det(A)\ne 0$, then $A$ is invertible and $A^{-1} = \frac 1{\det(A)} \operatorname{adj} A$
-1
votes
0answers
44 views

Handling the absolute value of a vector mathematically [on hold]

I am trying to do a mathematical analysis of something like the following Relu(x'A') Relu(Ax) = constant A here is also a circulant matrix. Relu changes its input element wise. It keeps the value ...
2
votes
3answers
42 views

Can you diagonalize the matrix $\mathrm P$?

Let be $P\in\mathbb{C}^{n\times n}$ so that $P^2=P$. The question is, when is $P$ diagonalizable? I got that the eigenvalues of $P$ can either be 0 or 1 and if there exists a diagonal matrix $D$ so ...
0
votes
0answers
33 views

When is the following trace inequality valid?

I have $A = A^T$ (and can have any real eigenvalue) and $B = B^T \succeq 0$ and want to know if the following holds $$ trace(AB) \leq 0 \iff \lambda_{max} (AB) \leq 0 $$ I know that the matrix $AB$ ...
1
vote
1answer
43 views

Show that the trace of A is less than n

Let $A$ be an $n\times n$ matrix with complex entries such that $A^k=I_n$ for some positive integer $k$. Show that the trace of $A$ satisfies $$|tr(A)| \leq n.$$ I have no idea how to approach this ...
7
votes
1answer
81 views

Proving $x\in\text{SL}(n,\mathbb Q)$ given finite indices of $x^{-1}Gx$ in $G$ and $x^{-1}Gx$

Denote $G=\text{SL}(n,\mathbb Z)$ and let $x\in \text{SL}(n,\mathbb R)$ such that $$[G:x^{-1}Gx\cap G],[x^{-1}Gx:x^{-1}Gx\cap G]<\infty.$$ Prove that $x\in\text{SL}(n,\mathbb Q)$. I know that $\...
1
vote
2answers
80 views

How come two of the eigenvalues are same?

Question is about finding the eigenvalues of the matrix : $$\begin{bmatrix} 0 & 0 & 2 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ \end{bmatrix}$$ the matrix would become $$\begin{bmatrix} -...
1
vote
0answers
38 views

Real and imaginary part of tensors of matrices

Given a matrix $A\in \mathbb{C}^{n\times m}$, clearly we can write $A=\Re(A)+i \Im(A)$, i.e., the real and imaginary part of $A$. (For instance, $A=[1,i]$, then $A=[1,0]+i[0,1]$). I am interested in ...
3
votes
3answers
77 views

Which non-negative matrices have negative eigenvalues?

It's easy to proof by counterexample that non-negative matrices can have negative eigenvalues. For example, the following matrix have -1 as an eigenvalue: $$ A = \begin{bmatrix} 0 & 0 & 0 ...
1
vote
1answer
33 views

Matrix differential equation of the form $X'=CX$

Let $n \in \mathbb{N}^{\ast}$ and $\mathrm{Sym}(n)$ (respectively $\mathrm{Spd}(n)$) denote the linear space (respectively set) of real $n \times n$ symmetric (respectively positive definite) matrices....
4
votes
2answers
52 views

Intuitive understanding of the matrix of a linear transformation

Is it accurate to say that a matrix $M(T)$ of the linear map $T:V\to W$ encodes the linear map into a series of numbers by showing how the linear map applied to the basis vectors of $V$ can be ...
2
votes
0answers
36 views

Relationship between eigenvalues of Hermetian matrices

Suppose that we have two $m\times m$ matrices $A$ and $B$ which are Hermetian, with $|B_{ij}|\leq |A_{ij}|$ for $i,j = 1,2 \cdots, m$. Can we say anything about the relationship between the largest ...
8
votes
1answer
88 views

Prove that matrix $A$ diagonalizable if $A^2=I$ using characteristic polynomial

Prove that the matrix $A$ is diagonalizable if $A^2=I$ using characteristic polynomial I saw an answer that used the minimal polynomial of $A$. Can that be proven without using minimal polynomial? ...
2
votes
3answers
48 views

A set of linear algebra questions?

Could you help me with these questions, I figured most of them out on my own, but I'm not completely sure if I'm correct. a) $A=\begin{bmatrix}a^2&ab&ac\\ ab&b^2&bc\\ ac&bc&c^...
3
votes
3answers
91 views

How to find the determinant of this $n \times n$ matrix in a clever way?

\begin{bmatrix} b_1 & b_2 & b_3 & \cdots & b_{n-1} & 0 \\ a_1 & 0 & 0 & \cdots & 0 & b_1 \\ 0 & a_2 & 0 & \cdots & 0 &...
1
vote
5answers
150 views

Good true-false linear algebra questions?

Can you suggest me a collection of true-false linear algebra questions, like the ones found in the MIT exams, if possible with solutions (i.e. explanations)? Sorry if it turns out that my request is ...
1
vote
0answers
19 views

numerical range of a parametrized matrix

I want to find the numerical range of a parametrized matrix $A(t)$ such that $(a_ij)=\exp (b_{ij}t)$ where $B=(b_{ij})$ is a real matrix. does anyone know something about the numerical range of a ...
2
votes
0answers
9 views

When do bounds for eigenvalues become strict?

Let $A$ be a real square matrix with eigenvalues $\lambda_k(A), \, k=1, \dots, n$. Further, let $S = (A+A^T)/2$ denote the symmetric part of $A$. Bendixson (1902) showed that $$ \min_j \lambda_j(S) \...
0
votes
0answers
33 views

Solve for a matrix “trapped in” summations?

I am trying to solve for matrix $X$ that is "trapped amid a summation." $$\sum_{i}^N\left(XA_iX^TXA_i^T+XA_i^TX^TXA_i\right)-\sum_{i}^M\left(B_iX+B^T_iX\right)=0,$$ assuming matrix dimensions match ...
1
vote
0answers
22 views

How to compute 2-norm between two matrices of different sizes?

I have a matrix $A$ of size $n\times8$ and matrix B of size $m\times8$. I need to compute $2$-norm to measure similarity. It can be measured as: $d(A,B)= \frac{||A-B||_{2}}{||A||_{2} ||B||_{2}}$
2
votes
2answers
54 views

Commutativity of matrix and its transpose

If a matrix is symmetric or skew-symmetric it commutes in the obvious way with its transpose. (For symmetric: $SS^T=S^2$ and $S^TS =S^2$) The less obvious is the case of commutativity for orthogonal ...
3
votes
2answers
446 views

Solving for the inverse of a matrix

I'm currently working on a problem in special relativity and it requires me to find the inverse of a matrix of a very specific form as follows: Given a 4x4 matrix $A$ of the form $$A = I + a\cdot\vec{...
1
vote
3answers
52 views

Derivative of trace of fourth-order matrix product?

I've been able to find all I need in the handy Matrix Cookbook, but today I encountered a form that is not in the book. Could anyone show me what is $$\frac{\operatorname{d}}{\operatorname{d}X}\...
0
votes
5answers
87 views

Why does this $6\times 6$ matrix has a null determinant?

about this matrix: $$ \left( \begin{array}{ c c c c c c} 1& 0& 0& \mathit{1}& 0& 0\\ 0.5& 0.5& 0& 0.5& \mathit{0.5}& 0 \\ 0.5& 0& 0....
22
votes
6answers
3k views

If I generate a random matrix what is the probability of it to be singular?

Just a random question which came to my mind while watching a linear algebra lecture online. The lecturer said that MATLAB always generates non-singular matrices. I wish to know that in the space of ...
2
votes
1answer
44 views

Matrices representing injective homomorphisms

Let $R$ be a ring and $M$, $N$ finitely generated free modules modules over $R$. Let $A$ be a matrix representing a homomorphism $f: M \rightarrow N$. We know that the map $f$ is injective if and only ...
2
votes
2answers
29 views

The lower bound of the smallest eigenvalue of a symmetric positive definite matrix

I encounter a symmetric positive definite matrix whose features are all diagonal entries are $1$. all the other entries are in $[0, 1)$, but the matrix is not diagonally dominant. Now I am ...
3
votes
2answers
56 views

Is orthogonality of column vectors preserved after right-multiplication by unitary matrix?

$\mathbf V$ is an $n \times (n-1)$ matrix with mutually orthogonal columns. $\mathbf Q$ is a unitary matrix of size $(n-1) \times (n-1)$. Is there a concise algebraic proof that the columns of $\...
0
votes
0answers
11 views

A matrix decomposition problem for row/column element order

Indeed, I don't know how to classify this problem, but I try to use matrix to describe it. The problem is that there exists a function $f(x, y)$ and its exact form remains unknown. But I have some ...
2
votes
1answer
35 views

Adjoint of a normal operator A is a polynomial in A

Is it true that adjoint of a normal operator A can be written as a polynomial in A?
2
votes
0answers
16 views

Calculating characteristic polynomial of linear transformation, cofactor expansion?

My question more is if cofactor expansion is really the best way to calculate the characteristic polynomial here: Linear transformation: $$T: P_2(R) -> P_2(R)$$ defined by $$T(a_o + a_1x + a_2x^2)...
1
vote
0answers
37 views

basic set of operation to transform square matrixes

There is any finite set of operations to transform any square matrix to any other square matrix (of the same order) ? If there isn't, can we construct a stable sub group with such property?
0
votes
0answers
20 views

Quantizing a matrix of reals while preserving row and column sums

Assume $E = [\epsilon_{i,j}]$, $i=1,2,\dotsc,m$, $j=1,2,\dotsc,n$, is an $m\times n$ matrix of reals. We know that $\forall i,j$, $\epsilon_{i,j}\in[-1/2,1/2]$. Moreover we know that both row-sums ...
0
votes
0answers
16 views

Optimize an Trace matrix form

In paper " Generalized Low Rank Approximations of Matrixces the Dimension of matrix are follow: $A_i$ is $r$ x $c$ L is $r$ x $l_1$ R is $c$ x $l2$ $D_i$ is $l_1$ x $l_2$ why it says ...
3
votes
2answers
43 views

Inverse of a matrix and a scalar

I'm asked o find $$\det((ad-bc)^{-1}\begin{bmatrix}a & b \\ c& d \end{bmatrix})$$ what I did was : This would equal to $$\det (\begin{bmatrix}\cfrac{1}{(ad-bc)} & 0 \\ 0 & \cfrac{1}...