Let V, W be two matrices of the same order. If all entries of V are nonzero, then we say that X is Schur invertible and define its Schur inverse, V(-), by V(-).V = J, where J is the matrix with all 1's. The vector space M₁ (F) of n x n matrices acts on itself in three distinct ways: if CE M₁ (F) we can define endomorphisms Xc, Ac and Yc by XCM := CM, ACM: C. M, Yc := MCT. Let A, B ben x n matrices. Assume that X₁ is invertible and AB is invertible in the sense of Schur. Note that XA is invertible if and only if A is, and AB is invertible if and only if the Schur inverse B(-) is defined. We say that (A, B) is a one-sided Jones pair if ΧΑΔΒΧΑ = ΔΒΧΑΔΒ· We call this the braid relation. Give an example for a one-sided Jones pair.

Let V, W be two matrices of the same order. If all entries of V are nonzero, then we say that X is Schur invertible and define its Schur inverse, V(-), by V(-).V = J, where J is the matrix with all 1's. The vector space M₁ (F) of n x n matrices acts on itself in three distinct ways: if CE M₁ (F) we can define endomorphisms Xc, Ac and Yc by XCM := CM, ACM: C. M, Yc := MCT. Let A, B ben x n matrices. Assume that X₁ is invertible and AB is invertible in the sense of Schur. Note that XA is invertible if and only if A is, and AB is invertible if and only if the Schur inverse B(-) is defined. We say that (A, B) is a one-sided Jones pair if ΧΑΔΒΧΑ = ΔΒΧΑΔΒ· We call this the braid relation. Give an example for a one-sided Jones pair.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 23EQ

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