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Products of orthogonal projections and polar decompositions
 Linear Algebra Appl
"... Abstract We characterize the sets X of all products P Q, and Y of all products P QP , where P, Q run over all orthogonal projections and we solve the problems arg min{ P − Q : (P, Q) ∈ Z}, for Z = X or Y. We also determine the polar decompositions and MoorePenrose pseudoinverses of elements of X. ..."
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Abstract We characterize the sets X of all products P Q, and Y of all products P QP , where P, Q run over all orthogonal projections and we solve the problems arg min{ P − Q : (P, Q) ∈ Z}, for Z = X or Y. We also determine the polar decompositions and MoorePenrose pseudoinverses of elements of X.
© Hindawi Publishing Corp. ON MIXEDTYPE REVERSEORDER LAWS FOR THE MOOREPENROSE INVERSE OF A MATRIX PRODUCT
, 2003
"... Some mixedtype reverseorder laws for the MoorePenrose inverse of a matrix product are established. Necessary and sufficient conditions for these laws to hold are found by the matrix rank method. Some applications and extensions of these reverseorder laws to the weighted MoorePenrose inverse are ..."
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Some mixedtype reverseorder laws for the MoorePenrose inverse of a matrix product are established. Necessary and sufficient conditions for these laws to hold are found by the matrix rank method. Some applications and extensions of these reverseorder laws to the weighted MoorePenrose inverse are also given. 2000 Mathematics Subject Classification: 15A03, 15A09. If A and B are a pair of invertible matrices of the same size, then the product AB is nonsingular, too, and the inverse of the product AB satisfies the reverseorder law (AB)−1 = B−1A−1. This law can be used to find the properties of (AB)−1, as well as to simplify various matrix expressions that involve the inverse of a matrix product. However, this formula cannot trivially be extended to the MoorePenrose inverse of matrix products. For a general m×n complex matrix A, the MoorePenrose inverse A† of A is the unique n×m matrix X that satisfies the following four Penrose equations: (i) AXA=A,