Eigenvalue and Characteristic Root I confused with two terminologies in my lecture, due to an expert. Statement on board I wrote is

Roots of characteristic polynomial of real orthogonal matrix have modulus $1$.

But the expert said, please (in fact, "you should") write

If $\lambda$ is an eigenvalue of a real orthogonal matrix, it should satisfy $|\lambda|=1$.

Which statements are more appropriate? Can anybody give me a good reference for standard definitions of "Eigenvalue"?

"$\lambda$ is an eigenvalue" means that there is a non-zero vector $v$ such that $Mv=\lambda v$. It is equivalent to saying that $M-\lambda I$ has determinant zero, which is equivalent to saying that $\lambda$ is a root of the characteristic polynomial.
The two statements you have given say the exact same thing, just the second one is more explicit (since it uses less words and more equations).
