A restatement of the major principal direction alignment principle for robustness quantification

The major principal direction alignment (MPDA) principle, developed in the context of robust control theory, states that a necessary and sufficient condition for the spectral radius of a matrix to equal its maximum singular value is that the major input and the major output principal directions of the matrix be aligned. The MPDA principle emerged from a study of the derivatives of the maximum singular value. An ambiguity that occurs when the maximum singular value is repeated is considered in this paper, together with a modified statement of the major principal direction alignment principle. The new necessary and sufficient condition for the spectral radius of a matrix to equal its maximum singular value is that there exists at least one major input and output principal direction pair of the matrix that is aligned. A rigorous proof is provided for the new necessary and sufficient condition, which makes use of early results on dual norms and dual vectors. An example is presented to illustrate the results.