HW10 - 462 MATRIX ANALYSIS Use MATLAB throughout to check...

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Unformatted text preview: 462 MATRIX ANALYSIS Use MATLAB throughout to check your answers. Show that the eigenvalues of the matrix —1 6 —12 A: 0 713 30 0 —9 20 are 5, 2 and —1. Obtain the corresponding eigenvectors. Write down the modal matrix M and spectral matrix A. Evaluate M’| and show that M’[AM= A. Usingthe eigenvalues and corresponding eigenvectors of the symmetric matrix 0 A: 0 3 obtained in Example 6.9, verify that MTAlfi': A where M and A are respectively a normalized modal matrix and a spectral matrix of A. Given 8~211 find its eigenvalues and corresponding eigenvectors. Normalize the eigenvectors and write down the cone-spending normalized modal matrix All. Write down MT and show that ”TAM : A, where A is the spectral matrix of A. Determine the eigenvalues and corresponding eigenvectors of the matrix 1 l —2 A = —1 2 1 0 l —1 Write down the modal matrix M and spectral matrix A Confirm that M"AM = A and that A = MAM—1. Determine the eigenvalues andu eigenvectors of the symmetric . 3 _2 4 A: —2 —2 6. ‘4 6 —1 Verify that the eigenveetors are . and write down an orthogonal LTAL = A, where A is the spectral A 3 X 3 symmetric matrix A has 6, 3 and l. The eigenvectors co to the eigenvalues 6 and l are [l [—2 1 0]T respecitively. Find th corresponding to the eigenvalue _- determine the matrix A. Given that l = l is a thriceerepea H ' of the matrix —3 —7 —5 A=2 4‘3 12 2 use the nullity, given by (6.11)1 of . matrix to show that there is only one? linearly independent eigenvector. i: finther generalized eigenvectors the corresponding modal matrix 7 that M"AM : J, where J is the an, Jordan matrix. 7 Show that the eigenvalues of the "- 1 0 —0.5 —3 .. *3 0 0 1 A: are —2, —2_, 4 and 4. Using the nu by (6.11), of appropriate matricest there are two linearly independent corresponding to the repeated Big and only one corresponding to the eigenvalue 4. Obtain a further : eigenvector corresponding to the Write down the Jordan'eanmlieal‘: 6.7 FUNCTIONS OF A MATRIX 481 3 1 2 compute A2 and, using the Cayley—Hamilton theorem, compute A7—3A“+A‘+3A3—2A2+31 'ts own characteristic equation. [1 fl Icy—Hamilton theorem to evaluate (1!) A3 (C) A“ Evaluate e“ for | 1 o 1 o (am—[l 1] (b)A=[1 2] Given '_ a ' tic equation ofan n x 11 matrix A is 2 O 0 ' +c,,_13.""+c,,,zl”'2+.t.+cll+co=0 A=g 0 1 1 0 0 l CayleyeHamjlton theorem, CHA'”I + CHAH + . . . + c,A + (:0! = 0 Show that 'u'gular then every eigenvalue is _ 0 d 0 0 0 5°C” an sinA2iA—12A2= 0 1 0 -. a 1 7‘ Tl: . —c—(A"+cfl_,AH+ . . .+ CIA) 0 0 1 .0 multiplying throughout by If1 gives Given (AH-I-CMA"‘2+.._+CJ) (6.44) A_ [3+1 2:4] 44) find the inverse of the matrix 5 -t r2 — t + 3 evaluate L1 2 7 ' _ (a) g (b) Ad: that the characteristic equation of the dr 1 Given A: t2+l 1—1 5 0 evaluate A2 and show that {4.)th l—‘HM ~7A~ll=0 d 2 (M and, using (6.44), determine A". (T: (A ) i 2‘“ E? ...
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