Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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4 views

Is Bs(1,-1) linear?

I would like to prove that the Baumslag-Solitar group $BS(1,-1)=\langle a,b| bab^{-1}=a^{-1}\rangle$ is embeddable in $GL_n(\mathbb{Z})$ for some nonnegative integer $n.$ So i tried to find two ...
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0answers
9 views

about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$

I am a little confused about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$. From other answers (Is a complex vector space closed under complex ...
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1answer
16 views

Minimal Polynomial

I have found the following characteristic polynomial: $$(x+2)(x-2)^2$$ I need to write down all the possible minimal polynomial, so I wrote: $${(x+2),(x-2),(x+2)(x-2),(x+2)(x-2)^2,(x-2)^2}$$ Why is ...
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0answers
6 views

Looking for references on the complexity of computation of a basis transformation matrix

I'm looking for some references on the complexity for the following kind of problem: Given two Basis $(a_1, ... ,a_n)$ and $(b_1, ..., b_n)$ of the $K(x)$-vector space $V$ I want to compute the ...
2
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1answer
34 views

$A^k = I$ implies diagonalizable? [duplicate]

If $A$ is a square complex matrix with $A^k = I$ (where $I$ is the identity matrix of the same size as $A$) for some positive integer $k$, does it follow that $A$ is diagonalizable?
3
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1answer
19 views

$\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has form $X \mapsto AXA^{-1}$.

As the title suggests, what is the easiest way to see that any $\mathbb{C}$-algebra automorphism of $M_N(\mathbb{C})$ has the form $X \mapsto AXA^{-1}$ for some fixed $A \in GL_n(\mathbb{C})$?
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1answer
18 views

Prove $2a^Tb \leq \|a \|^2 + \|b\|_*^2$ with dual norm

We know we can easily (**) prove $$ 2a^Tb \leq \|a \|_2^2 + \|b\|_2^2, \forall a,b $$ Is there a way to prove the following: $$ 2a^Tb \leq \|a \|^2 + \|b\|_*^2, \forall a,b $$ where ...
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0answers
11 views

norm on $\mathcal{B}(H)$

Given a Hilbert space $H$ and $a$ be a real numbers $\geq‎‎‎ 1$ , let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T ...
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2answers
33 views

Finding eigenvalues and eigenvectors of $2 \times 2$ matix

I having a few issues finding the eigenvectors for the following matrix: $$ \begin{bmatrix} -1 & -1\\ 0 & -2 \\ \end{bmatrix}$$ I calculated the eigenvalues to be ...
4
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1answer
16 views

Exist basis, simultaneously upper-triangular?

Let $A, B \in M_n(\mathbb{C})$ be such that $\text{rank}(AB - BA) \le 1$. Does there exist a basis of $\mathbb{C}^n$ with respect to which $A$ and $B$ are simultaneously upper-triangular?
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1answer
16 views

For a linear system, why is direction “stored” in the variables when considering it as linear equations, but in vectors when its as a vector equation?

Given an arbitrary system of equations, why is direction in space "stored" in the variables when considering the system as linear equations, but "stored" in vectors when considering the system as a ...
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1answer
22 views

Is the free abelian group of rank 2 linear?

Is the group $\mathbb{Z}^2$ linear? By linear I mean There is a injective homomorphism from $\mathbb{Z}^2$ to $GL_n(\mathbb{Z})$ for some nonnegative interger $n.$ I tried the following homomorphism ...
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2answers
23 views

$f$ is a differentiable map and compute $Df(A)(H)$.

Let $f : GL(n, \Bbb R) \to GL(n, \Bbb R)$ be defined by $f(A) = A^{-1}$ where derivative of the matrix $A$ exists. Then $f$ is a differentiable map and compute $Df(A)(H)$. $A A^{-1} = I \implies ...
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1answer
23 views

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$?

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$? If this is false in general, is it possibly true for nilpotent ...
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1answer
25 views

Peculiar family of apparently positive semidefinite matrices

Let $x_1, \dots, x_n > 0$ be positive real numbers. From numerical experiments, it appears that the $n \times n$ matrix $$A_{ij} = \frac{1}{x_i + x_j} $$ is always positive semidefinite. Is ...
2
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3answers
36 views

Basis for subspace in $\mathbb{R}^4$

How would I start to answer this: Show that the vectors $(1,0,0,1)$, $(0,1,0,1)$, and $(0,0,1,1)$ form a basis for the subspace $V$ of $\mathbb{R}^4$ which is defined by the equation ...
0
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1answer
21 views

$m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?

As the title suggests, how do I see that an $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?
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0answers
30 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
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1answer
31 views

linear algebra (norm) [on hold]

Can someone explain to me the following definition - $\|T\|$ := $ \sup \{\|T(v)\| : v \in \mathbb{R}^n, \|v\| = 1\}$ where $T$ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ and ...
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0answers
25 views

How do you find the null space of an inconsistent system? [on hold]

For example, the augmented matrix: $$\left(\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$
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0answers
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quadratic form associated with projection operator in Hilbert space

we are in Hilbert space $L^2 $ 1) we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
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1answer
23 views

Finding an explicit eigenvector

Let $A$ be an $n\times n$ matrix over a field and let $\operatorname{adj}(A)$ denote its classical adjoint. Suppose all column sums of $A$ are zero so that $A$ is singular. If $\operatorname{rank}(A) ...
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0answers
10 views

Geometrical interpretation of the condition number as measure of matrix dissimilarity

Consider two $p$ by $p$ symmetric positive definite matrices $\pmb F$ and $\pmb G$ and denote $$\pmb D=\pmb G^{-1/2}\pmb F \pmb G^{-1/2}.$$ Sometimes, the condition number of $\pmb D$ will be used ...
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2answers
46 views

Distinct eigenvalues and matrices problem

Let $V$ be a real vector space and $T: V \rightarrow V$ be a linear transformation. It is given that if $v_1, . . . , v_n$ are eigenvectors for distinct eigenvalues $λ_1, . . . λ_n$ then $\{v_1, . . ...
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0answers
27 views

A question about minimizing the $\lambda_{max}$ over a set of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ entry matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (if it helps you can assume that $(1)$ all the ...
4
votes
1answer
34 views

Simultaneous orthogonal diagonalization of two matrices

Let $A=\begin{pmatrix} 1 & -2\\ -2 & 5 \end{pmatrix}$ and $B=\begin{pmatrix} -3 & 6\\ 6 & -10 \end{pmatrix}$. Obviously $A$ is positive-definite and thus we can simultaneously ...
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0answers
13 views

Action of the Symplectic Group on Siegel Upper Half Plane

Given $G= \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in Sp_{2n}(\mathbb{R})$ one can define an action on the symmetric $n \times n$ complex matrices with positive definite imaginary part by ...
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5answers
86 views

Solve: $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$

It is asked to solve the ODE $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$ for $x(0)=10, \; x'(0)=0$ It is equivalent to the first order system in two variables $$\begin{bmatrix} x' \\ y' \end{bmatrix} = ...
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0answers
45 views

Good algebra book to cover these topics?

I will be studying two algebra modules next year and I am looking for a comprehensive book that will cover both of them, however due to having very minmal exprience with algebra I am looking for your ...
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0answers
29 views

The Intution Behind Real Symmetric Matrices and Their Real Eigenvectors

I am wondering about the geometric intuition behind real symmetric matrices and their corresponding linear transformations. Is it possible to understand geometrically why real symmetric matrices ...
1
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1answer
16 views

Finding the minimal polynomial of a linear operator

Let $P=\begin{pmatrix} i & 2\\ -1 & -i \end{pmatrix}$ and $T_P\colon M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ a linear map defined by $T_P(X)=P^{-1}XP$. I need to find the ...
1
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1answer
65 views

Matrix exponential of $\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix}$

It is asked to evaluate the matrix exponential of $$A=\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix}$$ It is not hard to do this, since this matrix have 3 ...
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0answers
32 views

Row rank$=$Column rank

This is one of the proofs given on Wikipedia. Let $A$ be an $m \times n$ matrix with entries in the real numbers whose row rank is $r$. Therefore, the dimension of the row space of $A$ is $r$. Let ...
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0answers
18 views

Distance/Similarity between matrices (different size) [on hold]

I have many matrices that have different size. Specifically, those matrices have the same number of rows but vary in the number of column. Each row is a different signal measurements, and each column ...
0
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1answer
11 views

Verifying expression with MP Pseudoinverse

Numerical simulations suggest that the expression $$ A=G^\dagger (PGG^\dagger P)^+G, $$ where $^+$ denotes the Moore-Pensore pseudoinverse, $P$ is the projector $$ P=I-\frac{1}{c^\dagger G^\dagger G ...
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0answers
25 views

Kernel of homomorphism

Let $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I wanna show that the following homomorphisms $f_1$ and $f_2$ defined by $f_1: H\to GL_n(\mathbb{Z})$ $f_1(p)=P$ and $f_1(q)=Q$ ...
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1answer
29 views

Find subsets $W$ and $V$ of $\mathbb{R}^3$ such that $\mathbb{R}(W\cap V)\neq\mathbb{R}W\cap \mathbb{R}V$.

Find subsets $W$ and $V$ of $\mathbb{R}^3$ such that $\mathbb{R}(W\cap V)\neq\mathbb{R}W\cap \mathbb{R}V$. I'm not sure how to find these sets. I'm sure there is an elementary solution. Any solutions ...
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2answers
23 views

Row sum of $P^{m}$ when row sum of $P$ is $1$

Let $P$ be an $n\times n$ matrix whose row sum equals $1$. Then for any positive integer $m$ , what is the row sum of $P^{m}$ ? Now I took arbitrary $2\times 2$ matrix ...
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0answers
12 views

Connected components of pseudospectra

In this Article, page 5 Theorem 2.3 ,what is connected components of pseudospectra of matrix polynomial?
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12 views

Matrix & Linear Algebra - Rows Expressed as Linear Combinations of a Set of Linearly Independent Vectors

The question arises from a proof for showing that matrices and their transposes have the same rank, in the textbook Advanced Engineering Mathematics by Erwin Kreyszig. A matrix of a certain size and ...
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1answer
27 views

Concavity of distance function in $\mathbb{R}^n$ or determinant of $(x^T \cdot x)$

I would like to compute the concavity of the distance function in $\mathbb{R}^n$. Let $ f(x) =- \Vert x \Vert $ in $\mathbb{R}^n$. Then $\nabla_xf=- \frac{x}{\Vert x \Vert}$. And ...
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0answers
38 views

Retrieve the value of x,z and x [on hold]

I want to learn about HOW to calculation in order to retrieve the value of x, y and x. Do you have a recommended tutorial to for a beginner in relation to linear algebra in this specific case? I ...
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0answers
11 views

on triangular matrices and inverses [duplicate]

Suppose we have $A$, an upper triangular matrix. Can we conclude that $A^{-1}$ must be upper triangular as well? $A$ non singular. I mean it seems obvious but how can we prove it? by induction?
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1answer
39 views

The Calculation Process

I don't understand HOW the calculation is done to retrieve the value 9, -6 and 18? Thanks!
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0answers
25 views

Eigen vectors of a matrix multiplied with its transpose [on hold]

Do the eigen vectors of $A A^T$ and $AA^T$ belong to the row, column, null or left null spaces of the matrix $A$?
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1answer
34 views

Nicest operators on a vector space

Axler writes in his book that "nicest operators on $V$ are those for which there is an orthonormal basis of $V$ w.r.t which the operator has a diagonal matrix". i.e. orthonormal basis of $V$ ...
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19 views

What is connected components of pseudospectra of matrix polynomial? . [on hold]

What is connected components of pseudospectra of matrix polynomial? Please see this link
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1answer
35 views

Sketching phase portrait of an ellipse

I have a system of linear ODE's as follows: $$\frac{dx}{dt} = y, \frac{dy}{dt} = -4x$$ which has solution $$\begin{bmatrix}x\\y\end{bmatrix} = \alpha\begin{bmatrix}\cos2t\\-2\sin2t\end{bmatrix} + ...
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1answer
16 views

Assigning a specific value to components of a vector

So far, I've run into this twice and I'm not exactly sure how to make this connection myself, but in this case, I've been asked to find the dot product of $(i+j+k) \cdot (3i+2j-5k)$ I understand ...
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0answers
27 views

How to prove this decomposition

There are two vectors l=$(l_1,l_2)^T$, m=$(m_1,m_2)^T$, and a symmetric matrix S=$\begin{bmatrix}s_{11}&s_{12}\\s_{12}&s_{22}\end{bmatrix}$. Then, ...