Linked Questions

1
vote
0answers
421 views

Commuting diagonalisable matrices are simultaneously diagonalizable [duplicate]

Possible Duplicate: Simultaneous diagonalization If $A$ and $B$ are diagonalisable matrices such that $AB=BA$, prove that there exists an invertible matrix $P$ such that $P^{-1}AP$ and $P^{-1}...
0
votes
1answer
72 views

Show that a matrix is diagonal [duplicate]

My task given by my professor was the following: $Let \,\, A, \,\, B \in \mathbb{C}^{n\times n}$ be selfadjoint and such that $[A, B] := AB-BA=0.$ Show that $C:= A+iB$ is normal. Show further that ...
2
votes
1answer
80 views

Problem about linear algebra [duplicate]

Suppose we have two $n \times n$ square matrices A and B such that $AB=BA$. It is known that A, B and AB all have n distinct eigenvectors that is a basis of $\mathbb{C}^n$. Can we then show that there ...
1
vote
0answers
56 views

Find a basis such that two linear transforms can be diagonalized at the same time [duplicate]

Possible Duplicate: Simultaneous diagonalization Assume $V$ is a n-dimension vector space over a number field $\Bbb{K}$, $\mathscr{A}$ and $\mathscr{B}$ are two linear transforms in $V$, and ...
1
vote
1answer
49 views

Show that if $RT = TR$ then $R$ and $T$ are given by diagonal matrices [duplicate]

Suppose we that $V$ is finite dimensional vector space over $\mathbb{C}$ and $R, T$ are diagonalizable operators such that $RT = TR$. Show that there exists basis of $V$ such that both $R$ and $T$ are ...
83
votes
1answer
54k views

Simultaneously Diagonalizable Proof

Two $n\times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices. a.) Show that ...
17
votes
3answers
5k views

If matrices $A$ and $B$ commute, $A$ with distinct eigenvalues, then $B$ is a polynomial in $A$

If $A\in M_{n}$ has $n$ distinct eigenvalues and if $A$ commutes with a given matrix $B\in M_{n}$, how can I show that $B$ is a polynomial in $A$ of degree at most $n-1$? I think first I need to show ...
6
votes
2answers
4k views

Proof of Spectral Theorem

There are a number of results called Spectral Theorems. This question deals with the Linear Algebra result on normal operators, which has the self-adjoint case as a particular case. In class, we saw ...
3
votes
2answers
688 views

Simultaneous diagonalization

Given two symmetric matrices $A,B\in\Bbb R^n$ how can we find if they are simultaneously diagonalizable? If they have such property how can we find $U$ such that $UAU'$ and $UBU'$ are simultaneously ...
9
votes
2answers
188 views

Let $A$ and $B$ in $O_n(\mathbb{R})$. Show that $A$ and $B$ commute.

Let $A$ and $B$ in $O_n(\mathbb{R})$ (orthogonal matrices) such that $|||B-I_n|||<\sqrt{2}$ (subordinate norm) and $A$ commute with $BAB^{-1}$. Show that $A$ and $B$ commute. My 'attempt': ...
0
votes
3answers
153 views

Matrices that Commute with of a Specific matrix

Let $a$ and $b$ be real numbers. Considere the $2\times 2$ matrix \begin{equation*}A=\left[\begin{array}{cc}a&b\\-b&a\end{array}\right]. \end{equation*} What is the centralizer of the matrix $...
4
votes
2answers
67 views

Suppose $A,B$ are positive operators with $AB=0$,what is the norm of $A+B$?

Suppose $A,B$ are positive operators with $AB=0$,what is the norm of $A+B$?
3
votes
1answer
183 views

Minimize $\operatorname{tr}(X^TA^TAX(X^T(I-P)X)^{-1})$ by solving an eigenproblem?

My optimization problem is $$\min_X\operatorname{tr}(X^TA^TAX(X^T(I-P)X)^{-1}),$$ where $P$ is a projection matrix. I was told this could be solved as an eigenproblem: columns of $X^*$ (the solution) ...
2
votes
2answers
99 views

Changing the spectrum of a matrix while preserving its eigenvectors.

Give a matrix $\textbf{A} \in \mathbb{C}^{n\times n}$ with eigenvalues $\lambda_1, ..., \lambda_n$ and eigenspaces $$ E_{\lambda_j} = \{\textbf{v} \in \mathbb{C}^n \mid \textbf{Av} = \lambda_j \...
0
votes
1answer
81 views

Matrix Algebras: Generator

Problem Given the algebra $\mathcal{M}_\mathbb{C}(2)$. Denote the normals: $$\mathcal{N}:=\{N\in\mathcal{M}_\mathbb{C}(2):N^*N=NN^*\}$$ And their calculus: $$\mathcal{N}(N):=\{\eta(N):\eta\in\...

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