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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text: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 C = M₁ (F) we can define endomorphisms Xc, Ac and Yc by
XCM := CM,
ACM: C. M, Yc := MCT.
Let A, B be n x n matrices. Assume that XÃ is invertible and AB is invertible
in the sense of Schur. Note that XÃ is invertible if and only if A is, and Aß 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 = ΔBΧΑΔΒ·
We call this the braid relation. Give an example for a one-sided Jones pair.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
