Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Eigenvalues of a Hermition Matrix do not cross

Wikipedia's article on avoided crossing asserts that "The eigenvalues of a Hermitian matrix depending on N continuous real parameters cannot cross except at a manifold of N-2 dimensions." If it's ...
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1answer
94 views

using fixed point theorem

Hi I want to use the fixed point theorem to show that for $G: \mathbb{R}^n \rightarrow \mathbb{R}^n$ $G(x)= \epsilon M x + \max(x,y)$, here $y$ is given and $\max(x,y)$ is the vector with component ...
2
votes
2answers
128 views

Linear Algebra: Proof of determinants

I need to show that if a and b are two non-zero members of Field q, then the number of nxn matrices with determinant a is the same as the number with determinant b. Im really not sure how how to ...
3
votes
4answers
186 views

Is there a good intuitive way to understand why matrix B is inverse of A when matrix A|I is turned into I|B

I'm looking for some help with my intuition of basic matrix operations, specifically finding a matrix's inverse (as per my subject line). I have no problems with the steps. The basic row operations ...
3
votes
2answers
202 views

Eigenvector of matrix of equal numbers

For matrix the matrix $$A = \begin{bmatrix} 3&1&1\\ 1&3&1\\ 1&1&3\\ \end{bmatrix}$$ with eigenvalues $\lambda_1=5$, $\lambda_2=2$, $\lambda_3=2$, I am trying to find the ...
3
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1answer
102 views

bilinear form $F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$, find ortogonal subspaces, that satisfy…

Define $F$ as bilinear form $M_n(\mathbb{R}) \text{ x } M_n(\mathbb{R}) \rightarrow \mathbb{R}$ $F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$ Prove, that $F$ is represented by ...
2
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2answers
68 views

little question about linear operators

Let H be a complex Hilbert Space. Let $P \in L(H)$ be an idempotent operator ($P^{2} = P$). Also, let $\parallel P\parallel = 1$. I want to prove that $P$ is an orthogonal operator. I defined $M = ...
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0answers
96 views

Diagonalizing the sum of a matrix and a multiple of the identity matrix

Suppose we have a matrix $A = B+\lambda I$, where $B\in \mathbb{R}^{n\times n}$, $I$ is the identity matrix and $\lambda\in \mathbb{R}$. If I know the eigenvalues and eigenvectors of $B$, what can I ...
0
votes
1answer
687 views

Condition number for non-square matrix?

From what I understand the condition number of a non-square matrix A is its largest singular value divided by its smallest nonzero singular value: $\kappa(A) = \sigma_1/\sigma_n $. Where ...
0
votes
2answers
128 views

A question about the determinant of matrices with integer entries

Motivated by some Physics backgrounds, let's consider the following group $GL_n(\mathbb Z)$ which consists of matrices satisfying some conditions. Let $M_n(\mathbb Z)$ be the set(not a group) of ...
1
vote
1answer
48 views

Wronskian of $y, y_1,\dots, y_n$

Let $y_1,\dots, y_n$ be linearly independent functions in $C^\infty$. For each $y \in C^\infty$, define $T(y) \in C^\infty$ by $$[T(y)](t)=\begin{vmatrix} y(t) & y_1(t) & \cdots & y_n(t)\\ ...
2
votes
1answer
31 views

Simplifying an expression with vectors in $\mathbb{R}^3$

Consider the vectors $u,v,w\in\mathbb{R}^3$, given by $u=(u_1,u_2,u_3)$, $v=(v_1,v_2,v_3)$ and $w=(w_1,w_2,w_3)$. I'm looking for a simplification for the following expression $$(\langle v,w\rangle ...
3
votes
1answer
290 views

Parameters to represent degrees of freedom in $n\times n$ orthogonal real matrices

An $n\times n$ orthogonal real matrix $A$ is a set ${A_{ij}}$ of $n^2$ real numbers that satisfy the constraints: $$\sum_k A_{ik} A_{kj} = \delta_{ij} $$ for all $1\leq i,j\leq n$. The equations ...
6
votes
0answers
71 views

Duality of $Z(G)$ and $[G,G]$ in representation?

This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group. I was thinking about its manifestation in group ...
4
votes
2answers
134 views

How can I prove $\det(\overline M)=\overline{\det(M)}$?

Of course $\overline M$ is the complex conjugate of an $n\times n$ matrix $M$. Someone gave me advice to use the definition of determinant, then it means I have to use cofactor expasion here?
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vote
4answers
94 views

Linear Algebra: proof

Show that $\alpha, \beta \in \mathbb R^n$ are orthogonal if, and only if, $\|\alpha-\beta\|=\|\alpha+\beta\|$ My first thought was to square each side as to get rid of the square root, and then its ...
0
votes
2answers
39 views

Matrix multiplication and eigen vectors

If $a$ is a right eigenvector of $S$ and $R^T$ with eigenvalue $1$. How would determine $a^TRSa$? Is $Sa$ simply $a$? Any hints that apply here would be greatly appreciated.
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0answers
131 views

Properties of eigenvectors

If a is a right eigenvector of Z and b is a right eigen vector of Y, is a * b' a right eigenvector of Z * Y?
3
votes
1answer
171 views

Representing linear operators on infinite sequences as infinite matrices

So this question arose while I was working on a homework assignment to find the adjoint operator of a continuous operator on $l^2$. Anyway, it seemed like I could find it if I thought about the ...
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2answers
49 views

Diagonalization of a matrix $A \in GL(n,\mathbb{C})$

I am trying to show that $$ \frac{d}{dt} \log \det A_t = Tr (A^{-1}_tA'_t) $$ where $A_t \in GL(n,\mathbb{C})$ and $A'_t = \frac{d}{dt}A_t$. I think I can show why this is the case if $A$ is ...
0
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1answer
93 views

Transforming linear equation over $GF(2)$ to system of the equations

I'm working on the problem of solving systems of linear equations over $GF(2)$ using SAT-solver. There is a one step in the algorithm that I don't clearly understand. During this step I need to ...
12
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1answer
418 views

An unusual type of linear algebra problem

I've come across the following linear algebra problem while trying to derive something in information theory. I'm looking both for numerical ways to solve this type of problem and for anything ...
1
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2answers
79 views

$T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$

Question Let $T:V \rightarrow V$ be a linear map. How do I prove that $T \circ T = T_0$ ( the zero linear map) iff $R(T) \subset N(T)$? Attempt \begin{eqnarray} T\circ ...
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2answers
167 views

find a solution from power series for multiple variable

$3^x4^y = 4,782,969 $ where $x$ and $y$ are integers. What is the value of $y$? Is there any theory to solve this type problem? i have tried to make $4,782,969$ into power series but couldn't. So a ...
2
votes
1answer
64 views

Necessary condition of $|\sum a_n|=\sum |a_n|$

Let $A\in M_{n\times n} (\mathbb{R})$ in which each entry is positive. Let $v$ be a nonzero vector in $\mathbb{C}^n$ where $v_k\neq 0$ Suppose $|\sum_{j=1}^n A_{kj} v_j|=\sum_{j=1}^n |A_{kj} v_j|$. ...
1
vote
1answer
43 views

I need help constructing a $n\times n$ matrix with determinant $1$ that moves the unit vector $e_{1}=col(1,0,0,..0)$ to a non-zero vector $x$.

I need help constructing a $n\times n$ matrix with determinant $1$ that moves the unit vector $e_{1}=(1,0,0,..0)^T$ to a non-zero vector $x$ in $X=R^n\setminus\{0\}$. I need to use it to prove that a ...
1
vote
3answers
113 views

Find $\dim R(T)$ and $\dim N(T)$ from the matrix of a linear map $T$

Let \begin{eqnarray} \begin{pmatrix} 1&1&1\\ 1&1&1\\ 1&1&1 \end{pmatrix} \end{eqnarray} be the matrix of a linear map $T$. Find $\dim R(T)$ and $\dim N(T)$. My beginning ...
7
votes
1answer
208 views

When are two commuting linear operators functions of each other

I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up. If we formally consider the integral operators ...
0
votes
2answers
153 views

Show that if $\langle X,AX\rangle = 0$, then $AX = 0$

Let $A$ be an $n\times n$ symmetric positive semidefinite matrix. Let $X \in \mathbb{R}^n$ Show that if $\langle X, AX\rangle = 0,$ then $AX = 0$. This seems like it's really simple, but I must be ...
2
votes
2answers
110 views

If $\operatorname{rk}(A) = 1$, when $B + A$ is invertible?

I should know how do this problem, but I have troubles with it. Let $B$ be an invertible matrix and let $A$ be a matrix with $\operatorname{rk}(A) = 1$. Then $\exists \lambda$ such that $A^2 = ...
1
vote
1answer
50 views

some inclusions regarding linear operators

Let $H$ be a Hilbert Space and $T:H\rightarrow H$ a linear operator. Let $T^*$ be the adjoint operator of $T$ and let $\operatorname{Cl}(X)$ be the topological closure of the set X and $X^{\perp}$ ...
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1answer
61 views

Finding three independent solutions for the system

I'm stuck on this assignment. Not sure how to begin. Let $\begin {bmatrix}2&1&1\\ 0&2&3\\ 0&0&2\end {bmatrix} = A$ as part of the system $Ax=x'$. Find three independent ...
3
votes
1answer
102 views

Toric Varieties from Cones

Consider the lattice $N=\Bbb{Z}^d$ spanned by $e_1,\dots,e_d$ and the cone $$\sigma=\text{Cone}\{e_1,\dots,e_k\}, \quad k<d.$$ I am trying to understand why the toric variety $V_\sigma$ obtained is ...
7
votes
0answers
343 views

Monotone matrix norms

[Ciarlet 2.2-10] Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if ...
2
votes
2answers
46 views

Collecting Matrices when solving laplace transform

\begin{align} \tag{1} \dot x&=Ax\\ \tag{2} sX(s)-x(0)&=AX(s) \\ \tag{3} (sI-A)X(s)&=x(0) \end{align} Considering these equations, how can we go from $(2)$ to $(3)$? My question is why ...
2
votes
1answer
101 views

Formulate optimization problem

My research area has "nothing to do with mathematics" but I still find it full of optimization problems. Therefore, I would like to learn to formulate and solve such problems, even though I am not ...
2
votes
0answers
51 views

Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation} d_{\alpha}|\#G, \end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
0
votes
1answer
114 views

Sum of the Hyperharmonic\Over-harmonic Series under $\mathbb{Z}_n$ for $p=2$

For $n \geq 5$ prime number, calculate the sum of: $$1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{(n-1)^2}$$ under $\mathbb{Z}_n$. I figured it's the hyperharmonic\over-harmonic series, $$ ...
4
votes
1answer
107 views

To prove that the dimension of $V$ is $d_1^2 + \ldots + d_k^2$

Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $$(x - c_1)^{d_1} \cdots (x - c_k)^{d_k} , $$ where $c_1,\ldots,c_k$ are distinct. Let $V$ be the space of $n \times n$ ...
3
votes
5answers
77 views

On any finite field, adding the identity element a finite amount of times will result to the neutral element

Show that if you add the identity element ($1$) a finite amount of times will result to the neutral element ($0$). I started with saying that in every field there's an element $a \in F$ so that $a + ...
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vote
1answer
54 views

What is the right answer to $P^{-1}AP$?

Given the matrices $$A=\pmatrix{b+8c & 2c-2b & 4b-4c \\ 4c-4a & c+8b & 2a-2c \\ 2b-2a & 4a-4b & a+8b \\ }, P=\pmatrix{0 & 1 & 2 \\ 2 & 0 & 1 \\1 & 2 ...
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vote
1answer
46 views

$B=A(I+F)$ with $\|F\|<1$ implies $\|A^{-1}B\|<1/(1-\|B^{-1}A-I\|)$?

Let $A$ be invertible $n\times n$ matrix and $B=A(I+F)$ with $\|F\|<1$ where norm is consistent and submultiplicative. I learned that $I+F$ is invertible and $\|(I+F)^{-1}\|<1/(1-\|F\|)$. So we ...
1
vote
1answer
87 views

Inequality involving trace and operator norm

Here's a simple question for which I can't find an answer. Let $W$ be a square real matrix with eigenvalues all real and positive ($W$ is not necessarily symmetric nor diagonalizable) and $A$ a real ...
5
votes
2answers
235 views

Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: $g(1) = \alpha;$ $g(2n+j) = 3g(n) + γn + β_j$ : j = 0,1 and n >= 1 I have assumed the closed form to ...
3
votes
2answers
83 views

Is this determinant bounded?

Let $D_n$ be the determinant of the $n-1$ by $n-1$ matrix such that the main diagonal entries are $3,4,5,\cdots,n+1$ and other entries being $1$. i.e. $$D_n= \det \begin{pmatrix} ...
0
votes
1answer
60 views

decomposition of m-cycle in m-1 transpositions

I am searching for a proof. Every m-cycle $\sigma = (x_1 x_2 ... x_m)$ can be expressed as an composition of m-1 transpositions. I found many formulas, for example: $\sigma = (x_1 x_2)(x_2 x_3) ... ...
0
votes
1answer
95 views

Prove $T ^ {n} = T_ {1} + … + T_ {k}$

Let $ V $ complex inner product space (real) with $\dim V = n$ and let $T $ normal nonzero operator (symmetric) on $V$. Show that there are $k$ operators $T_1, ... T_k: k \le n$ on $V$ such that $T_i ...
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vote
2answers
383 views

irreducible, diagonally dominant matrix

I am facing a problem for irreducible,diagonally dominant matrices. How to prove that irreducible, diagonally dominant matrix is invertible? Please help me in this problem.
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3answers
132 views

Need Help with Proof please help! [closed]

Im not sure really how to do this, will someone please help. Given that $A$ and $B$ are $n\times n$ matrices over field $F$, prove that the solution set $S$ to the matrix equation $XA+BX=0$ is a ...
3
votes
1answer
89 views

For any matrix norm, is it true $||A|| \le \max|a_{ij}|\cdot ||(1)||$?

Let $|| \cdot ||$ be a matrix norm on $m \times n$ matrices, which is not assumed to be submultiplicative. Is it true that $||A|| \le \max|a_{ij}|\cdot ||(1)||$ where $(1)$ denotes the matrix with all ...