Consider the following block matrix: N a aT c where N is a m ×m real symmetric matrix, a is a m-dimensional real vector, and c is a real scalar. Prove that the following two statements are equivalent. i) M is a positive definite (PD) matrix. ii) N is PD and c > a⊤N −1a

Consider the following block matrix: N a aT c where N is a m ×m real symmetric matrix, a is a m-dimensional real vector, and c is a real scalar. Prove that the following two statements are equivalent. i) M is a positive definite (PD) matrix. ii) N is PD and c > a⊤N −1a

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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Question

. Consider the following block matrix:

N	a
a^T	c

where N is a m ×m real symmetric matrix, a is a m-dimensional real vector,
and c is a real scalar. Prove that the following two statements are equivalent.
i) M is a positive definite (PD) matrix.
ii) N is PD and c > a^⊤N ⁻¹a.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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