Laplacian and the Adjacency Matrices

Proof. We first recall that every non-singular matrix B can be written B = QR, where Q is an orthonormal matrix Q and R is upper-triangular matrix R with positive diagonals1 We will use a slight variation of this fact, writing B = RQ. Now, since QT = Q−1, QAQT has exactly the same eigenvalues as A. Let Rt be the matrix t ∗R+ (1− t)I, and consider the family of matrices Mt = RtQAQR t , as t goes from 0 to 1. At t = 0, the matrix has the same eigenvalues as A. At t = 1, we get BTAB. All of these matrices are symmetric, so they all have Real eigenvalues. As the eigenvalues of a This is called the QR-factorization. It follows from Gram-Schmidt orthonormalization.