Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
6. Generalization of Positive Definiteness E There is a generalization of the notion of positive (semi-)definitenes to non-Hermitian matrices (that we do not consider in this course). Any complex matrix A € Cdxd is said to be positive definite if R((x, Ax)) > 0, for all non-zero vectors x ≤ Cd, where R(c) denotes the real part of the complex number c E C. Positive semi- definiteness is also defined in the usual way. If the matrix A is real, i.e., A € Rdxd, the positive definiteness condition simplifies to having (x, Ax) > 0, for all non-zero real vectors x € Rª. E E Show that A € Rdxd is positive definite if and only if the symmetric part of A, i.e., (A + AT)/2, is positive definite in the widely used sense that we studied in the lectures.

6. Generalization of Positive Definiteness E There is a generalization of the notion of positive (semi-)definitenes to non-Hermitian matrices (that we do not consider in this course). Any complex matrix A € Cdxd is said to be positive definite if R((x, Ax)) > 0, for all non-zero vectors x ≤ Cd, where R(c) denotes the real part of the complex number c E C. Positive semi- definiteness is also defined in the usual way. If the matrix A is real, i.e., A € Rdxd, the positive definiteness condition simplifies to having (x, Ax) > 0, for all non-zero real vectors x € Rª. E E Show that A € Rdxd is positive definite if and only if the symmetric part of A, i.e., (A + AT)/2, is positive definite in the widely used sense that we studied in the lectures.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
6. Generalization of Positive Definiteness E There is a generalization of the notion of positive (semi-)definiteness to non-Hermitian matrices (that we do not consider in this course). Any complex matrix A € Cdxd is said to be positive definite if R((x, Ax)) > 0, for all non-zero vectors x = Cd, where R(c) denotes the real part of the complex number c E C. Positive semi- definiteness is also defined in the usual way. If the matrix A is real, i.e., A ¤ Rª×d, the positive definiteness condition simplifies to having (x, Ax) > 0, for all non-zero real vectors x € Rd. Show that A € Rdxd is positive definite if and only if the symmetric part of A, i.e., (A + AT)/2, is positive definite in the widely used sense that we studied in the lectures.
Transcribed Image Text:6. Generalization of Positive Definiteness E There is a generalization of the notion of positive (semi-)definiteness to non-Hermitian matrices (that we do not consider in this course). Any complex matrix A € Cdxd is said to be positive definite if R((x, Ax)) > 0, for all non-zero vectors x = Cd, where R(c) denotes the real part of the complex number c E C. Positive semi- definiteness is also defined in the usual way. If the matrix A is real, i.e., A ¤ Rª×d, the positive definiteness condition simplifies to having (x, Ax) > 0, for all non-zero real vectors x € Rd. Show that A € Rdxd is positive definite if and only if the symmetric part of A, i.e., (A + AT)/2, is positive definite in the widely used sense that we studied in the lectures.
Expert Solution
steps

Step by step

Solved in 3 steps with 38 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS