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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 conespending 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
(AHICMA"‘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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 Fall '10
 ChinC.Lee

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