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Martin Sleziak
Martin Sleziak
1:39 PM
For instructions how to render MathJax(TeX) in chat see this post on meta. Or have a look at bookmarklets on robjohn's website.
2
Q: Eigenvalue and similar matrices

Cloud JRif $A$ and $B$ are two $n\times n$ matrices with same eigenvalue having same algebraic and geometric multiplicity. Does $A$ and $B$ are similar? If $A$ is diagnalizable then the claim is true. But does it true even when sum of geometric multiplicity is not $n$. Please give me a hint to start w...

linear-algebra matrices eigenvalues-eigenvectors examples-counterexamples
5
A: Eigenvalue and similar matrices

Saucy O'PathNo, it's famously false. The usual counterexample is $A=\begin{pmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{pmatrix}, B=\begin{pmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}$ for both of which $0$ has algebraic multiplicity $4$ and geometric multiplicity $2$. The result that hol...

0
A: Eigenvalue and similar matrices

Miguel Botocounter example: $$\begin{bmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \end{bmatrix}$$ $$\begin{bmatrix} -1 & 1 & 0 & 0\\ 0 & -1 & 1 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\\ \end{bmatrix}$$ these two matrixes have the same eigen values and same geometric multipli...

Can you prove those matrices are non similar — Cloud JR 45 mins ago
You can first check the validity of these two observations.
Fact 1. If $A$, and $B$ are similar, so are $A+cI$, $B+cI$ for any constant $c$.
Fact 2. If $A$ and $B$ are similar, so are $A^k$ and $B^k$ for every positive integer $k$.
In fact, in a very similar way we could show that $p(A)$ and $p(B)$ are similar for any polynomial $p(x)$. (Although this is not needed here.)
From the first observation, it should be clear that the two answers are basically the same - their difference is just identity matrix.
For $$
A=\begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}
\qquad\text{and}\qquad
B=\begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}
$$ we have $$
A^2=0
\qquad\text{and}\qquad
B^2=\begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}.
$$
Now we can see that $A^2$ and $B^2$ are not similar. (Since only zero matrix is similar to $0$.)
Consequently, $A$ and $B$ are not similar.
@CloudJR I left a few comments on some possible arguments that they are not similar in linear algebra chatroom. — Martin Sleziak 1 min ago
 

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