Page 133 - Numerical methods for chemical engineering

Eigenvector matrix decomposition and basis sets 119

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Proof The eigenvalues of B are roots of det(B − λI) = 0. Substituting B = S AS,
det(B − λI) = det(S −1 AS − λI) = 0 (3.94)
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We now use the identity I = S S = S IS to obtain
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det(B − λI) = det(S −1 AS − λS −1 IS) = det(S (A − λI)S) = 0 (3.95)
Now, as det(AB) = det(A) × det(B),
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det(B − λI) = det(S ) × det(A − λI) × det(S) = 0 (3.96)
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As det(S) = 0 and det(S ) = 0 for a nonsingular S, λ can only satisfy det(B − λI) = 0if
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it also satisfies det(A − λI) = 0. Thus, A and B = S AS have the same eigenvalues.
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Let Aw = λw. Substituting A = SBS ,wehave SBS −1 w = λw. Multiplying by S −1
yields the following relationship between eigenvectors of A and B:
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Aw = λw B(S w) = λ(S w) (3.97)
QED

H
Definition The Hermitian conjugate of A, A , is obtained by taking the transpose of the
matrix and its complex conjugate,
H
A kj = a kj + ib kj {a kj, b kj }∈ (A ) kj = a jk − ib jk (3.98)
H
H
H
The Hermitian conjugate of a matrix product is (AB) = B A .
H
Definition A matrix A is said to be Hermitian if A = A . A real Hermitian matrix A is
T
symmetric, A = A .
H
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Definition A matrix U is said to be unitary if U = U . A real unitary matrix Q is
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T
orthogonal, Q = Q .
H
H
Definition A matrix A is said to be normal if AA = A A. Hermitian and unitary matrices
are both special cases of normal matrices.
Theorem EVG4 We can write any complex matrix A in terms of a unitary matrix U and
upper triangular matrix R as the Schur decomposition,
A = URU H (3.99)
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H
As U = U , A and R are similar, and as R is triangular, the diagonal elements of R are
eigenvalues of A. In MATLAB, a Schur decomposition is computed by schur.
Proof We proceed by induction, showing that if (3.99) holds for (N − 1) × (N − 1) matrices,
then it also does for N × N matrices. For N = 1, we have the trivial result

U = 1 A = R = λ (3.100)

[2]
For an N × N matrix A, let Aw = λw, |w|= 1, and let {w, u ,..., u [N] } form an