A polar representation of singular matrices
نویسندگان
چکیده
منابع مشابه
A Polar Representation of Singular Matrices
Let A = (aty), (i = l, 2, • • • , ra; j = l , 2, • • • , » ) , be a matrix of m rows and n columns, whose elements a,-,are complex numbers. I t has been shown f that , if m—n and A is non-singular, A=PiU= UP2, where U is a unitary matrix, while P\ and P 2 are positive definite hermitian matrices. Moreover in such a polar representation of A, as it has been called, the matrices Pi , P2, and U ar...
متن کاملA Generalization of the Polar Representation of Nonsingular Matrices
where P is a positive definite hermitian matrix and U is a unitary matrix. In this polar representation of the matrix A, as it is called, the two matrices P and U are unique. Since the matrix P is positive definite and nonsingular, it has the same signature as the identity matrix E while the unitary matrix U is a conjunctive automorph of E. From (1) we may deduce a somewhat similar representati...
متن کاملSingular values of convex functions of matrices
Let $A_{i},B_{i},X_{i},i=1,dots,m,$ be $n$-by-$n$ matrices such that $sum_{i=1}^{m}leftvert A_{i}rightvert ^{2}$ and $sum_{i=1}^{m}leftvert B_{i}rightvert ^{2}$ are nonzero matrices and each $X_{i}$ is positive semidefinite. It is shown that if $f$ is a nonnegative increasing convex function on $left[ 0,infty right) $ satisfying $fleft( 0right) =0 $, then $$2s_{j}left( fleft( fra...
متن کاملSingular value inequalities for positive semidefinite matrices
In this note, we obtain some singular values inequalities for positive semidefinite matrices by using block matrix technique. Our results are similar to some inequalities shown by Bhatia and Kittaneh in [Linear Algebra Appl. 308 (2000) 203-211] and [Linear Algebra Appl. 428 (2008) 2177-2191].
متن کاملMore about measures and Jacobians of singular random matrices
In this work are studied the Jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1935
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1935-06032-x