# In a Hermitian Matrix, the Eigenvectors of Different Eigenvalues are Orthogonal

This is an elementary (yet important) fact in matrix analysis.

Statement

Let $M$ be an $n\times n$ complex Hermitian matrix which means $M=M^*$ where $*$ denotes the conjugate transpose operation. Let $\lambda_1, \neq \lambda_2$ be two different  eigenvalues of $M$. Let $x, y$ be the two eigenvectors of $M$ corresponding to the two eigenvalues $\lambda_1$ and $\lambda_2$, respectively.

Then the following is true:

$\boxed{\lambda_1 \neq \lambda_2 \Longrightarrow \langle x, y \rangle = 0.}$

Here $\langle x,y \rangle := y^*x$ denotes the usual inner product of two vectors $x,y$.

# Eigenvalues of the Laplacian Matrix of the Complete Graph

We will show that the eigenvalues of the $n\times n$ Laplacian matrix $L$ of the complete graph $K_n$ are $\{0,n\}$ where  the eigenvalue $0$ has (algebraic) multiplicity $1$ and the eigenvalue $n$ has multiplicity $n-1$.

# Eigenvalues of a Hermitian Matrix are Real

In this post, we prove the following:

Statement: All eigenvalues of a Hermitian matrix are real.

Proof: