The Numerical Method For The Multivariate Eigenvalue Problem

The multivariate eigenvalue problem has its origins in the determination of canonical correlation coefficients for multivariate statistics. Its history goes back to when Hotelling(1936) first studied the so-called maximal correlation problem, which later was developed into what is knows as canonical correlation analysis. Applying the method of Lagrange multipliers such an optimization problem can be reduced to the multivariate eigenvalue problem (MEP) .Perturbation analysis plays an important role in modern numerical linear algebra. Backward errors reveal the stability of a numerical method. Condition numbers explain the sensitivity of the solution of a problem to backward error,the product of condition number times backward error provides a first order error bound for the computed solution. To our knowledge, however, the perturbation analysis for the MEP remains to be studied. This paper is mainly devoted to the perturbation analysis for the multivariate eigenvalue problem (MEP).In the second part, the concept of simple eigenvalue of a matrix is generalized to the MEP. Then, we study the first order perturbation expansions of a simple multivariate eigenvalue and the corresponding eigenvector. Consequently, we derive the explicit expressions of condition numbers, perturbation upper bounds and backward errors for multivariate eigenpairs. Finally, in the last part, we suggest a strategy of choosing the starting point for Horst's method and the P-SOR method. Numerical examples demonstrate that this strategy is natural and work well.