116 d e t h exercises for section 71 711 determine

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Unformatted text preview: . -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Gerschgorin circles derived from Gr ∩ Gc . In other words, one eigenvalue is in the circle centered at −5, while the other two eigenvalues are in the union of the other two circles in Figure 7.1.3. This is √ corroborated by computing σ (A)= {5, (1±5 5)/2} ≈ {5, 6.0902, −5.0902}. Example 7.1.6 Diagonally Dominant Matrices Revisited. Recall from Example 4.3.3 on p. 184 that An×n is said to be diagonally dominant (some authors say strictly diagonally dominant) whenever Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 500 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 n |aii | > |aij | for each i = 1, 2, . . . , n. j =1 j =i It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Gerschgorin’s theorem (7.1.13) guarantees that diagonally dominant matrices cannot possess a zero eigenvalue. But 0 ∈ σ (A) if and only if A is nonsingular / (Exercise 7.1.6), so Gerschgorin’s theorem provides an alternative to the argument used in Example 4.3.3 to prove that all diagonally dominant matrices are 68 nonsingular . For example, the 3 × 3 matrix A in Example 7.1.5 is diagonally dominant, and thus A is nonsingular. Even when a matrix is not diagonally dominant, Gerschgorin estimates still may be useful in determining whether or not the matrix is nonsingular simply by observing if zero is excluded from σ (A) based on the configuration of the Gerschgorin circles given in (7.1.16). D E T H Exercises for section 7.1 7.1.1. Determine the eigenvalues and eigenvectors for the following matrices. ⎛ ⎞ ⎛ ⎞ 2 16 8 3 −2 5 −10 −7 A= . B=⎝ 4 14 8⎠. C = ⎝0 1 4⎠. 14 11 −8 −32 −18 0 −1 5 ⎛ ⎞ ⎛ ⎞ 063 300 D = ⎝ −1 5 1 ⎠ . E = ⎝ 0 3 0 ⎠ . −1 2 4 003 IG R Y P Which, if any, are deficient in eigenvectors in the sense that there fails to exist a complete linearly...
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