Abstract: In this paper we showed that for an orthogonal projection matrix with rank $k$, there exists an principal submatrix with order $k$ of the matrix, such that its Rayleigh quotient has a positive lower bound. The proof was made by using the relation of the singular values, eigenvalues and norm of matrices, as well as the properties of unitary matrix and compound matrix.

Key words: orthogonal projection matrix, singular value, unitary matrix, compound matrix