5
votes
7answers
175 views

If $A^2 = B^2$, then $A=B$ or $A=-B$

Let $A_{n\times n},B_{n\times n}$ be square matrices with $n \geq 2$. If $A^2 = B^2$, then $A=B$ or $A=-B$. This is wrong but I don't see why. Do you have any counterexample?
2
votes
1answer
67 views

An operator such that $\|A\|^2 \neq \|A^2\|$

The question asks for a bounded linear operator on a Hilbert space satisfying the condition in the title. This is what I came up with: Let $A_1:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a 90-degree ...
6
votes
1answer
92 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
2
votes
1answer
39 views

Product of two symmetric banded matrices - real eigenvalues?

Let $A$ and $B$ be real symmetric banded matrices but $AB$ is not symmetric. Are the eigenvalues of $AB$ real? A more specific case: let $D$ be a real diagonal matrix, $B$ real symmetric and banded, ...
10
votes
1answer
283 views

Example of similar matrices $A$ and $B$ such that products $AB$ and $BA$ are not similar

I'm looking for the simplest possible example of square matrices $A$ and $B$ such that $A$ is similar to $B$, $AB$ is not similar to $BA$. Such an example should exist, but I would like to find ...
0
votes
0answers
22 views

is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
6
votes
1answer
66 views

Inner Product on a Vector Space over a field besides $\mathbb R$ or $\mathbb C$?

Are there any fields with vector spaces you can define an inner product over besides subfields of $\mathbb C$? I know that you'd want the field to contain an ordered subfield, so it must have ...
1
vote
1answer
57 views

Where can I find good examples about Algebra (but not only): Usual counter-examples, but also limit cases, rare ones, etc [duplicate]

I recently discovered the importance of examples and couter-examples in mathematics. Where could I find good examples books or anything related to it ? I am particularly looking for rare limit-cases, ...
0
votes
1answer
26 views

Hermitian matrices and their eigenvalues

Let $C=A+B$ where $A$ and $B$ are two hermitian matrices can I prove that $\lambda_{i,C}=\lambda_{i,A}+\lambda_{i,B}$ iff $x_{i,A}=x_{i,B}$? Where $x_i$ is the eigenvector related to eigenvalue ...
3
votes
3answers
100 views

Necessity of completeness of the inner product space in Riesz representation theorem

I wanted to find a counter example to show that the completeness of the inner product space is necessary in Riesz representation theorem. Please give an example of a bounded linear functional $T$ on ...
2
votes
4answers
78 views

Orthonormal basis of $\mathbb{R}^n$ containing $v_1 = \frac{1}{\sqrt{n}}(1,1,\ldots,1)$

I would like to construct the orthonormal basis $\{v_1,v_2,\ldots,v_n\}$ of $\mathbb{R}^n$ with $v_1 = \frac{1}{\sqrt{n}}(1,1,\ldots,1)$. I am looking for an analytic formulation of all vectors of the ...
-1
votes
3answers
103 views

Linear algebra. Find a counter-example

this is the statement: if $\vec v_{1}, \vec v_{2} , \vec v_{3}, \vec v_{4}$ is a basis for the vector space $\Bbb R^{4} $, and W is a subspace of $\Bbb R^{4}$, then some subset of the $\vec v$ 's is a ...
0
votes
2answers
82 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
0
votes
3answers
121 views

A finite dimensional vector space that is not naturally isomorphic to its dual.

I need an example of a finite dimensional vector space $V$ that is not naturally isomorphic to $V^\ast$. I know that, at least in finite dimensional case, there is a one-to-one correspondence between ...
0
votes
1answer
197 views

State space representation of transfer function $ 1/s $

I have three questions here: 1) I'm looking for a method to compute state-space equations by hand when given a transfer function, so far I think the best I've found is here. 2) Here's an example ...
0
votes
1answer
36 views

If $V$ is finite-dimensional and exist $\beta$ basis of $V$ such that $T(\beta)$ is a basis for $W$, then $T$ is a isomorphism?

Let $V$ and $W$ vector space over $F$ and $T : V \rightarrow W$ lineal. The statement is false, but I can't find a counterexample.
2
votes
1answer
117 views

Basic example of system controllability

Ok so I'm having a tough time understanding an example in my book, there is a great deal of hand-waving and I'm having a hard time following. I'd like to see the example worked out if possible, with ...
13
votes
6answers
315 views

Other guises for the vector space $\mathbb{R}^n$?

One way the vector space $\mathbb{R}^n$ can come up is as the space of polynomials over $\mathbb{R}$ of degree at most $(n-1)$ . Here we have the isomorphism: $$(a_0,a_1,\ldots,a_{n-1}) ...
0
votes
1answer
41 views

seek a special matrix

I am seeking the following example, maybe it is easy to construct, but I have no idea now besides aimless computation. Could anyone give me a matrix $A\in SL_n(\mathbb{Z})$ for some $n$, such that ...
2
votes
4answers
350 views

How to come up with a counter example in linear algebra

This came up in a problem I was working on. Problem:Let $V$ be an $n$ dimensional vector space over a field $F$. Let $T:V\rightarrow V$ be a linear operator and let $W$ be a $T$ invariant subspace of ...
2
votes
2answers
113 views

Trace zero matrix that can't be written as $AB - BA$?

According to this paper, every trace zero matrix over a field can be written in the form $AB - BA$. However, here's a basic counterexample: Let $A = diag(a, -a)$ for some nonzero number a. Then $A ...
4
votes
1answer
264 views

Tensor products over field do not commute with inverse limits?

In the question: Inverse limit of modules and tensor product, Matt E gives an example where inverse limits and tensor products do not commute over the base ring $\mathbb{Z}$. He then goes on to show ...
1
vote
1answer
79 views

Invariant Subspace Counterexample

Can someone give an example: Suppose $T \in L(V)$. If $V = W \bigoplus W'$ and if $W$ is T-invariant then $W'$ is not necessarily T-invariant.
1
vote
1answer
709 views

Example of a Markov chain transition matrix that is not diagonalizable?

It is well-known that every detailed-balance Markov chain has a diagonalizable transition matrix. I am looking for an example of a Markov chain whose transition matrix is not diagonalizable. That is: ...
4
votes
1answer
229 views

$A$ square matrix,nonsingular $\implies $ all submatrixes of $A$ are also nonsingular?

If a square matrix $A$ is nonsingular, then every submatrix of $A$ is also nonsingular. I am trying to come up with a counter example. But most involve really difficult examples, so I am starting to ...
10
votes
1answer
163 views

Other Euler characteristics?

At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, ...
1
vote
3answers
1k views

5 linear equations in 5 unknowns

I need an example of 5 linearly independent equations with 5 variables. How can I write such a equation set. As an example: ...
10
votes
3answers
261 views

Deducing results in linear algebra from results in commutative algebra

Here are two examples of results which can be deduced from commutative algebra: Any $n\times n$ complex matrix is conjugate to a Jordan canonical matrix (can be proven using the structure theorem ...
0
votes
1answer
260 views

Prove Axiom $10$ (Vector Spaces) independent of the others [duplicate]

Possible Duplicate: Is it possible to construct a quasi-vectorial space without an identity element? In Apostol Multivariable Calculus, $1.5$ exercise $30 b$, he asks the reader to prove ...
4
votes
2answers
157 views

The Duality Functor in Linear Algebra

I'm trying to gain an intuitive understanding of the following construction: For any vector space $M$ over a field $R$, one can define the algebraic dual of $M$ as $M^* := \mathsf{Hom}(M, R)$ and ...
1
vote
2answers
278 views

Example of eigenvectors in different bases (follow-up question)

This is a follow-up question on this one: Connection between eigenvalues and eigenvectors of a matrix in different bases Assume I have matrix $$ B=\left( \begin{array}{cccc} 0 & 0 & 1 ...
6
votes
3answers
3k views

Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...
9
votes
6answers
4k views

How are eigenvectors/eigenvalues and differential equations connected?

In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts. I learned that ...
5
votes
2answers
352 views

Tips for writing math solutions for others

I am working a bit on a collection of Linear Algebra examples, as well as some examples on induction. This is what is taught freshman year at our university. I intend to release this to the public, ...