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 \iff \langle x, y \rangle = 0.}

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

Continue reading “In a Hermitian Matrix, the Eigenvectors of Different Eigenvalues are Orthogonal”

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Eigenvalues of the Laplacian Matrix of the Complete Graph

611px-Complete_graph_K7.svg
A complete graph (source: David Benbennick)

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

Continue reading “Eigenvalues of the Laplacian Matrix of the Complete Graph”