Fractional Hadamard powers of positive semidefinite matrices

✍ Scribed by P. Fischer; J.D. Stegeman

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 197 KB
• Volume: 371
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(03)00421-x

✦ Synopsis

We consider the class S n of all real positive semidefinite n × n matrices, and the subclass S + n of all A ∈ S n with non-negative entries. For a positive, non-integer number α and some A ∈ S + n , when will the fractional Hadamard power A ♦α again belong to S + n ? It is known that, for a specific α, this holds for all A ∈ S + n if and only if α > n -2. Now let A ∈ S + n be of the form A = T + V , where T ∈ S + n has rank 1 and V ∈ S n has rank p 1. If the Hadamard quotient of T and V is Hadamard independent ('in general position') and V has 'sufficently small' entries, then a complete answer is given, depending on n, p, and α. Special attention is given to the case that p = 1.