Matrix and it's Transpose have the same eigenvalues?

Question

I have the following question in my book for mathematics: "Show that $A\in M_{n\times n}$has the same eigenvalues as $A^\tau$.

I can see how to do this for a $1\times 1,\ 2\times 2$, etc.

I would just always calculate the characteristic polynomial $p(\lambda)$, where $p(\lambda) = \det (A - \lambda I)$ and than solve for $\lambda$.

Now I can't really wrap my head around how to do this for a general $n\times n$ matrix. I think this would probably involve induction over $n$, but I'm not really sure how and how to get even started.

How does the characteristic polynomial look for the transposed matrix? — harfe, May 20 '20 at 08:08
Any square matrix is similar to its transpose and similar matrices have the same characteristic polynomial. — PinkyWay, May 20 '20 at 08:29
The problem is already solved thanks to Anurag A. , I will take a look at the suggested question later. — Matthias K., May 20 '20 at 08:30
@Matthias, it has already been solved before, you only have to search. It is your duty to see the "suggested" question. — PinkyWay, May 20 '20 at 08:31
Elegant proofs that similar matrices have the same characteristic polynomial?. Furthermore: A matrix and its transpose have the same set of eigenvalues. — PinkyWay, May 20 '20 at 08:32

score 2 · Accepted Answer · edited May 20 '20 at 09:43

2

\begin{align*} \det\left(A^T-cI\right) & = \det\left(A^T-cI^T\right) &(\because I^T=I)\\ & =\det(A-cI)^T\\ &=\det(A-cI) & (\because \det(M)=\det\left(M^T\right)) \end{align*} So the characteristic polynomials are same.

edited May 20 '20 at 09:43

PinkyWay

4,565

answered May 20 '20 at 08:11

Anurag A

41,067

2

Could you also explain what the three dots are, I have never seen them before? – Matthias K. May 20 '20 at 08:12
@MatthiasK. It means "since" or "because" – Anurag A May 20 '20 at 08:12
Oh ok, than this explains it very well and easy, thank you! – Matthias K. May 20 '20 at 08:13
1

@AnuragA This question has already been answered by @Fretty – PinkyWay May 20 '20 at 08:44

Matrix and it's Transpose have the same eigenvalues?

1 Answers1