Question

If A is real-symmetric, it has real eigenvalues. What can you say about the eigenvalues if A is real and
anti-symmetric (A = ..AT )? Give both a general explanation for any nn A (similar to what we did
in class and in the book) and check by finding the eigenvalues a 22 anti-symmetric example matrix.

Answer 1

This is where i got this quaction http://web.mit.edu/18.06/www/Spring09/pset8-s09.pdf

Answer 2

suppose \(A^t=-A\in\mathbb{R}^{n\times n}\), i.e. a anti-symmetric matrix with real entries.

Answer 3

let \(\lambda\) be an eigenvalue of \(A\) with corresponding eigenvector \(x\), then
\[ \large Ax=\lambda x \]

Answer 4

from this
\[ \large x^tAx=x^t(\lambda x)=\lambda(x^tx) \]
\[ \large \lambda=\frac{x^tAx}{x^tx}\in\mathbb{R} \]

Answer 5

Really thanks.......!!!

Answer 6

u r welcome