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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0answers
44 views

Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?

(Roughly related, but generalizing, of this earlier question) Background: The first part of the following(the column-wise-focus) is also described in Eri Jabotinski's 1953-treatize Representation of ...
1
vote
1answer
35 views

Determinant of Hermitian Tridiagonal Matrix with Constant Upper and Lower Diagonals

I got this equation where the a terms are known but I want to determine a relationship between the b terms (so, no numerical analysis please). I know that the bi terms are real and the a terms are ...
0
votes
1answer
25 views

Next step to show that these matrice expressions are equal?

This is a problem from Discrete Mathematics and its Applications I know invertible means it is possible to take the inverse of this matrix. This is definition of a power of a square matrix from my ...
0
votes
0answers
28 views

Determinant of matrix n x n [duplicate]

How to calculate $det\begin{bmatrix}1 & x_1 & x_1^2 \dots x_1^{n-1} \\ 1 & x_2 & x_2^2 \dots x_2^{n-1} \\ \\ 1 & x_n & x_n^2 \dots x_n^{n-1}\end{bmatrix}$?
1
vote
1answer
35 views

the relation between positive definite matrix and its ellipsoid

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Then, we associate with each $A \in S^n_{++}$ ...
2
votes
1answer
18 views

$B - A \in S^n_{++}$ and $I - A^{1/2}B^{-1}A^{1/2} \in S^n_{++}$ equivalent?

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Now suppose that $A,B \in S^n_{++}$ are two ...
0
votes
0answers
17 views

Positive/Negative Definite Bordered Hessian?

I understand how to check a function for concavity and convexity using the Hessian matrix and the rules for the determinants of the leading principal minors. I understand if these rules are violated, ...
1
vote
0answers
34 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
0
votes
1answer
19 views

Rewrite an expression in terms of basis vectors

Given any vector k $\epsilon$ $R^{3}$ consider k= $\sum_{j=1}^{3}$ $c_{j}u_{j}$ where $u_{1}$,$u_{2}$,$u_{3}$ are the orthonormal basis vectors (I don't know how to make them bold sorry about that, ...
4
votes
2answers
85 views

Finite cyclic subgroups of $GL_{2} (\mathbb{Z})$ [duplicate]

How could we prove that any element of $GL_2(\mathbb{Z})$ of finite order has order 1, 2, 3, 4, or 6? I am aware of the proof supplied here at this link: ...
1
vote
1answer
223 views

calculating the determinant of an $n \times n$ integer matrix

I want to write a polynomial algorithm for calculating the determinant of an $n \times n$ integer matrix. There are various codes in different programming languages on the web but unfortunately I am ...
1
vote
2answers
26 views

limit of a function with a matrix exponential

I spent too many time trying to solve this problem...and finals are coming. Please help me! I just can't see a method to do this demonstration: "For an $A_{n \times n}$ matrix, demonstrate that a ...
2
votes
1answer
44 views

proof on similarity of matrices

Could you please help me with the following problem? Let $A$ be an $n$$\times$$n$ complex matrix. Prove that $A$ is similar to $B$, which is an $n$ $\times$ $n$ real matrix, if and only if $A$ is ...
0
votes
1answer
25 views

One eigenvalue and eigensystem

Matrix $A \in \mathbb{K}^{n,n}$ has one engenvalue $\lambda \in \mathbb{K}$ and its engensystem $V_{\lambda}$ has dimension that equals to $n$. How to show that $A = \lambda I_{n}$?
0
votes
1answer
30 views

Converting Quaternion or 4x4 Matrix to 3x3 Matrix representation.

I'm working on some code that manipulates an Axis-Aligned Bounding Box, so it always encompasses the object it borders. I use a 3x3 matrix to re-size the box when it rotates. The only issue is I only ...
0
votes
1answer
22 views

Rayleigh quotient inequality. ($|\lambda-q^*Aq|\leqslant2||A||_{2}||v-q||_{2}$ )

Given $A\in\mathbb{C}^{n\times n}$ and $(\lambda,v)$ an eigenpair verifying $\|v\|_{2}=1$, $q\in\mathbb{C}^n$ a unitary vector. Show that its Rayleigh quotient verifies ...
0
votes
1answer
30 views

If two matrices have the same characteristic polynomials, determinant and trace, are they similar?

If two $n \times n$ matrices have the same characteristic polynomials, determinant and trace, are they similar, EVEN if ($ \lnot \#Spec= 0$)?
2
votes
2answers
29 views

How to prove this result?

Let {$\Delta_1,\Delta_2,\Delta_3\cdots\cdots\cdots\cdots\Delta_n$} be the set of all determinants of order 3 that can be made with the distinct real numbers from set $S=\{1,2,3,4,5,6,7,8,9\}$. Then ...
-4
votes
2answers
44 views

Determine the matrices $A$ and $B$ if $2A + 3B$ and $A + B^T $ is given

If $$ 2A + 3B = \begin{bmatrix} 8 & 3 \\ 7 & 6 \end{bmatrix} $$ and $$ A + B^T = \begin{bmatrix} 3 & 1 \\ 3 & 3 \end{bmatrix} $$ Determine the $2\times 2$ matrices $A$ and $B$.
1
vote
0answers
15 views

Applications of Matrix in simplifying algebra [closed]

Inversions (and the Mobius Transformation, though it belongs to complex numbers) are pretty good tools in simplifying an algebric mess. What other tools exist apart from this and how may we use them? ...
3
votes
1answer
60 views

Induction proof about entries of powers of strictly upper triangular matrix

Let $A$ be a $n \times n$ strictly upper triangular matrix. Prove that, for $k \ge1$, the matrix $A^k$ has the property that $(A^k)_{i,j} = 0$ for all $(i,j)$ with $j-i < k$. Also, show that $A^n ...
0
votes
1answer
20 views

How to find the transition matrix from basis $E$ to $E'$

Suppose there is a linear transformation $T$ on $\mathbb R^n$. And $$E=[\epsilon_1,\epsilon_2...\epsilon_n]$$and $$E'=[\epsilon'_1,\epsilon'_2,...\epsilon'_n]$$ are two different basis of $\mathbb ...
2
votes
1answer
54 views

Example of a non singular square matrix such that $A+A^{-1} = 0$

Is there any example of a non singular square matrix $A$ such that $A+A^{-1} = 0$? Are they any specific type of matrices or can these be found under any category of matrices (such as symmetric, ...
0
votes
2answers
61 views

Proof matrices and their eigenvalues

Let $C=A-B$ where $A=\begin{bmatrix}I &0\\ 0&0\end{bmatrix}$ and $B$ is a Laplacian matrix of a connected graph, so sum for rows is null and it doesn't have any zero row(or column). ...
1
vote
2answers
91 views

Laplacian matrix eigenvalues

Let $L$ be the laplacian of a connected graph. Is the maximum eigenvalue of $A=\begin{bmatrix} I & 0\\0&0\end{bmatrix} -L$ different than $1$??
2
votes
1answer
52 views

the solution of matrix polynomials

In order to get the eigenvalues of \begin{equation} P=\left[ \begin{array}{cc} 0_{n\times n} & I_{n\times n} \\ -A & -B% \end{array} \right], \end{equation} where $A$ and $B$ are both $n\times ...
4
votes
1answer
50 views

What is the Laplacian Matrix used for?

You can turn graphs into several matrix forms depending on what data you want to focus on. Does the Laplacian form have any uses on its own, or does it need to be paired with other things as some ...
4
votes
0answers
101 views

diagonal of pseudoinverse of laplacian matrix

I have to find the diagonal of the pseudoinverse of a laplacian matrix evaluated on a directed and weighted graph. My laplacian is defined as: L = D - A where: D is a diagonal matrix; Di,i the sum ...
2
votes
0answers
24 views

median eigenvalue

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...
0
votes
0answers
63 views

Spectral moments of signless Laplacians through eigenvalues of the line graph?

For a simple graph G, the following relationships hold: $$RR^T=\Delta+A$$ and $$R^TR=2I+A_{L(G)}$$ where R is the incidence matrix, A is the adjacency matrix, I is the identity matrix, $A_{L(G)}$ is ...
1
vote
0answers
144 views

Extracting hitting times from the pseudoinverse of a Laplacian matrix for an undirected graph [closed]

Provided a pseudoinverted Laplacian matrix for an undirected graph $G$, how can I extract first passage and commute times between vertex pairs in $G$?
2
votes
0answers
23 views

Problems in metric space including matrices. [closed]

Let $M(n, \Bbb R)$ denote the set of a real $n \times n$ matrices. We can always define a linear isomorphism between $M(n, \Bbb R)$ and $\Bbb R^{n^2}$....where the isomorphism is defined as for any ...
8
votes
1answer
118 views

nilpotent elements of $M_2(\mathbb{R})$, $M_2(\mathbb{Z}/4\mathbb{Z})$

Let $R$ be a commutative ring. We define $\mathfrak{N}(R)$ to be the set of nilpotent elements in $R$. Find $\mathfrak{N}(R)$ for: $R = M_2(\mathbb{R})$ $R = M_2(\mathbb{Z}/4\mathbb{Z})$
1
vote
2answers
39 views

A projection $P$ is orthogonal if and only if its spectral norm is 1

I have to show what the title says. A projection $P$ is orthogonal if and only if its spectral norm is $1$. I suppose I have to use the following identity: ...
1
vote
1answer
31 views

Matrix multiplication computation

Any tips how to solve this? $$ \left[ \begin{matrix}1 & 2 & 0 \\ -2 & -5 & 1 \\ 11&15&5 \end{matrix}\right] \times \mathbf{X} \times \left[ \begin{matrix} -4&5&1\\ ...
1
vote
1answer
18 views

Vector notation question

Just a short question regarding notation: If this matrix represents a vector and I want to solve it for $t=2$, may I write it as follows: $ \left( \begin{array}{ccc} vt\\ vt-gt\\ \end{array} ...
1
vote
0answers
28 views

Good resource to learn geometric interpretations of matrices [duplicate]

I need a good resource to learn matrices and all its properties through geometry. I feel geometry gives an insight into many matrix operations and a good resource will be useful for many students who ...
0
votes
0answers
7 views

Composition of a rotation and a homothetic transformation of different centers?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$. Furthermore let $h_{\lambda,S}$ be the homothetic transformation of center $S\neq \Omega$ and ratio $\lambda$. What ...
2
votes
1answer
52 views

Do linear operators $A$, $B$ satisfying $A = B+BAB$ commute?

I have two linear continuous operators $A$, $B$ on Banach space $X$ (for example, square matrices), satisfying the equation $$ A = B + BAB, $$ and such that the continuous inverses $(\mathrm{Id} ...
0
votes
1answer
46 views

symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix eigenvalues

I am trying to show that the symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix have corresponding eigenvalues $\lambda_i$ and $1 - \lambda_i$ for i=1 to n. $\lambda$ is ...
3
votes
0answers
195 views

Low-rank approximation to the Graph Laplacian matrix of a regular grid.

As mentioned in the title, does anybody know any methods of efficient low-rank approximation $LL^T$ to the Graph Laplacian matrix $A$ corresponding to a square lattice? (except PCA)
0
votes
1answer
91 views

Reading a Laplacian Matrix and its labeled graph?

How can the following labeled graph be extracted from the Laplacian Matrix below and viceversa? I had a look at this great conversation but it is already too advanced for me.
1
vote
1answer
55 views

Rank-one modification of graph Laplacian

Suppose I have a Laplacain matrix for a 3-node-path graph as follows $L=\left[\begin{array}{ccc} 1 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1 \end{array}\right]$ Now, I want to ...
4
votes
1answer
107 views

What graph Laplacians commute

I know that the graph Laplacian of a fully connected graph commutes with the Laplacian of any other graph. Is there any theorem stating something similar about some more general family of graphs? ...
4
votes
3answers
165 views

Nipotent matrix over a ring

This question is linked to this one: nilpotent elements of $M_2(\mathbb{R})$, $M_2(\mathbb{Z}/4\mathbb{Z})$ Let $R$ be a commutative ring with unity and let $A\in M_2(R)$. Show that $A$ is a ...
0
votes
0answers
40 views

Eigen vectors of graph laplacians

I have been reading about spectral graph theory from Daniel A. Spielman's notes. Fiedler’s Nodal Domain Theorem from this note says that : Let $G = (V, E, w)$ be a weighted connected graph, and let ...
3
votes
1answer
66 views

Determining $\det(\mathbf{A})$ using the characteristic polynomial

Let the 3x3 matrix be $ \mathbf{A} = \begin {bmatrix} 3&1&0\\1&3&0\\0&0&1 \end {bmatrix}$. a) Determine its eigenvalues and eigenvectors. b) Do the eigenvectors ...
2
votes
0answers
19 views

Transformations invariant wrt. $L_1$ norm.

$A$ is a real matrix of size $n \times k$, where $k \leq n$. $A$ has independent columns. Characterize the class of matrices $M \in \mathbb{R}^{k \times k}$ such that: $\forall x \in \mathbb{R}^k.\; ...
0
votes
2answers
113 views

Proving that $\det (A^2 - I) < 0 \Rightarrow \lambda \in (-1,1)$

Let $A$ be real square matrix. If $\det (A^2 - I) < 0$, then $A$ has an eigenvalue $\lambda \in (-1,1)$. How to prove this?
5
votes
1answer
27 views

How do I show that $\inf\limits_{\det(X)\neq0}\|X^{-1}AX\|^{2}_{F}=\sum\limits_{\lambda\in{\Lambda}}|\lambda|^{2}$?

Show that $$\inf\limits_{\det(X)\neq 0}\|X^{-1}AX\|^2_F=\sum_{\lambda\in\Lambda}|\lambda|^{2}$$ holds, where $\Lambda(A)$ is the set containing all eigenvalues of A, and $\|\cdot\|_{F}$ is the ...