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.

Given : xTCx > 0 , C∈ℝn×n , Ck is nonsingular . To Prove : yTCky > 0 , y∈ℝk

Answered: Let CE Rxn be a matrix such that xCx >…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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