Lanczos algorithm explained in statistics

The Lanczos algorithm is a well-known three-term recurrence that can be used to generate an orthogonal basis for a Krylov subspace derived by a symmetric matrix. In the paper, we present a statistical interpretation of the entries of the tridiagonal matrix generated by the Lanczos process with a diagonal matrix X and an initial vector e. We show that the entries on the main diagonal line can be interpreted as weighted mean and the entries on the super-diagonal line can be understood as weighted sum of variance. Besides, a recurrence for producing the entries on the off-diagonal entries of the tridiagonal matrix is discovered, which leads to a new implementation of the Lanczos process. Finally, numerical examples are provided to investigate the preservation of orthogonality and efficiency in data fitting.