{'fraction_non_alphanumeric': 0.019402985074626865, 'fraction_numerical': 0.0, 'mean_word_length': 4.545454545454546, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Stability and convex hulls of matrix powers

ABSTRACT Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e. being a P-matrix). These results are extended by considering convex combinations of higher powers of A and of their rows. The invertibility of matrices in these convex hulls is associated with the eigenvalues of A lying in open sectors of the right-half plane and provides a general context for the theory of matrices with P-matrix powers.