Matrices that have same complex eigenvalues with the same algebraic multiplicities have same trace?

Question

Question: Matrices that have same complex eigenvalues with the same algebraic multiplicities have same trace?

I was unable to find counter example

(For simplicity, I was searching for two $2×2$ matrices with real entries that same complex eigenvalues with same algebraic multiplicities but having different trace. So I have considered

$A=\begin{bmatrix}-2&-1\\5&2\end{bmatrix}$ clearly $A$ has complex eigenvalues $i, -i$ & both have algebraic multiplicity $1$. But I was unable to find matrix $B$ that have same eigenvalues with same algebraic multiplicities but having different trace )

Is the above statement is true?

Please help.

Trace is the sum of the eigen values, so there is no such example. — geetha290krm, Oct 23 '22 at 05:30

score 1 · Accepted Answer · answered Oct 23 '22 at 06:34

1

It is true, the trace of a matrix is equal to th3 sum of its eigenvalues (algebraic multiplicities included).

Proof that the trace of a matrix is the sum of its eigenvalues

answered Oct 23 '22 at 06:34

Marija

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Matrices that have same complex eigenvalues with the same algebraic multiplicities have same trace?

1 Answers1