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# Com 566 chapter 7 eigenvalues and eigenvectors

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Unformatted text preview: 1.html It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] 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,...
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