Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Let CE Rxn be a matrix such that xCx > 0 for all nonzero vectors & Prove that for k= 1,...,n, the leading principal submatrix C, satisfies y all nonzero vectors y R. Consequently, each C is nonsingular. R" Cky > 0 for

Let CE Rxn be a matrix such that xCx > 0 for all nonzero vectors & Prove that for k= 1,...,n, the leading principal submatrix C, satisfies y all nonzero vectors y R. Consequently, each C is nonsingular. R" Cky > 0 for

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
Let \( C \in \mathbb{R}^{n \times n} \) be a matrix such that \( \mathbf{x}^T C \mathbf{x} > 0 \) for all nonzero vectors \( \mathbf{x} \in \mathbb{R}^n \). **Prove** that for \( k = 1, \ldots, n \), the leading principal submatrix \( C_k \) satisfies \( \mathbf{y}^T C_k \mathbf{y} > 0 \) for all nonzero vectors \( \mathbf{y} \in \mathbb{R}^k \). Consequently, each \( C_k \) is nonsingular.
Transcribed Image Text:Let \( C \in \mathbb{R}^{n \times n} \) be a matrix such that \( \mathbf{x}^T C \mathbf{x} > 0 \) for all nonzero vectors \( \mathbf{x} \in \mathbb{R}^n \). **Prove** that for \( k = 1, \ldots, n \), the leading principal submatrix \( C_k \) satisfies \( \mathbf{y}^T C_k \mathbf{y} > 0 \) for all nonzero vectors \( \mathbf{y} \in \mathbb{R}^k \). Consequently, each \( C_k \) is nonsingular.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
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