Com 566 chapter 7 eigenvalues and eigenvectors

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1.html It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Buy from AMAZON.com 7.6 Positive Definite Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 561 equation can be solved in isolation. To extract solutions, the equations must somehow be uncoupled, and here’s where matrix diagonalization works its magic. Write equations (7.6.1) in matrix form as ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛y ⎞ 2 −1 y1 0 1 ⎜ −1 ⎟⎜ y2 ⎟ ⎜ 0 ⎟ 2 −1 ⎜ y2 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜⎟ .. ⎟⎜ y3 ⎟ ⎜ 0 ⎟ ⎜ y3 ⎟ + T ⎜ . −1 2 ⎟⎜ ⎟ = ⎜ ⎟, or y + Ay = 0, ⎜ . ⎟ mL ⎜ ⎜ ⎟⎝ . ⎠ ⎝ . ⎠ .. .. ⎝.⎠ . ⎝ . . . . . −1 ⎠ . 0 yn yn −1 2 (7.6.2) with y(0) = c = (c1 c2 · · · cn )T and y (0) = 0. Since A is symmetric, there is an orthogonal matrix P such that PT AP = D = diag (λ1 , λ2 , . . . , λn ), where the λi ’s are the eigenvalues of A. By making the substitution y = Pz (or, equivalently, by changing the coordinate system), (7.6.2) is transformed into ⎞⎛ ⎞ ⎛ ⎞ ⎛z ⎞ ⎛ λ1 0 · · · 0 z1 0 1 z + Dz = 0, ⎜ z2 ⎟ ⎜ 0 λ2 · · · 0 ⎟ ⎜ z2 ⎟ ⎜ 0 ⎟ ˜ z(0) = PT c = c, or ⎜ . ⎟ + ⎜ . . .. . ⎟⎜ . ⎟ = ⎜ . ⎟. ⎝.⎠ ⎝. . . . ⎠⎝ . ⎠ ⎝ . ⎠ . . . . . . z (0) = 0, 0 0 0 · · · λn zn zn D E T H IG R In other words, by changing to a coordinate system defined by a complete set of orthonormal eigenvectors for A, the original system (7.6.2) is completely uncoupled so that each equation zk + λk zk = 0 with zk (0) = ck and zk (0) = 0 can be ˜ solved independently. This helps reveal why diagonalizability is a fundamentally important concept. Recall from elementary differential equations that Y P √ + βk e−t −λk √ √ αk cos t λk + βk sin t λk zk + λk zk = 0 =⇒ zk (t) = O C √ αk et −λk when λk < 0, when λk ≥ 0. Vibrating beads suggest sinusoidal solutions, so we expect each λk > 0. In other words,...
View Full Document

This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online