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Unformatted text preview: ___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University  Ansan 41 Chapter 4. Eigenvalues and eigenvectors ♣ Numerical properties of a matrix 1. Eigenvalues & eigenvectors Definition Let A be an ) ( n n × square matrix. A scalar λ is called an eigenvalue of A if there is a nonzero vector x s.t. Ax = λ x . Such a vector is called the eigenvector of A corresponding to λ . ← Eigenvalue and eigenvector are always defined in pairwise, so that we denote ( λ , x ) as an eigenpair of A . ← An eigenvalue can be interpreted as the natural frequency or normal mode of the system. Definition Let A be an ) ( n n × square matrix and λ be an eigenvalue of A . The collection of all eigenvectors corresponding to λ , together with the zero vector, is called the eigenspace of λ and is denoted E λ . ← The set of all eigenvectors corresponding to an eigenvalue λ of A is just the set of nonzero vectors in the null space of A λ I . 2. Determinants (1) We define the determinant of a ) 1 1 ( × matrix [ ] a = A to be, a a = =   ) det( A . (2) The determinant of a ) 2 2 ( × matrix ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 22 21 12 11 a a a a A is 21 12 22 11   ) det( a a a a − = = A A (3) The determinant of a ) 3 3 ( × matrix ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 33 32 31 23 22 21 13 12 11 a a a a a a a a a A is 32 31 22 21 13 33 31 23 21 12 33 32 23 22 11   ) det( a a a a a a a a a a a a a a a + − = = A A For example, for ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − = 3 1 2 2 1 2 3 5 A , 5 ) det( = A ___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University  Ansan 42 Permutation Suppose we want to arrange a set of integers { 1, 2, …, n } in some order without repetition. This is the idea of permutation. We know that there are n!way of arrangements. ▶ Denote a permutation of { 1, 2, …, n } by { j 1 , j 2 , …, j n }. An inversion is said to occur in a permutation { j 1 , j 2 , …, j n } whenever a larger number precedes a smaller one; e.g. for a permutation { 6, 1, 3, 4, 5, 2 }, there are 5+0+1+1+1=8 inversions. ▶ A permutation is called even if the total number of inversions is even and odd, if odd. ▶ The elementary product from a matrix ) ( n n × ∈ A is any product of n entries from A , none of which come from the same row and column; e.g. ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 22 21 12 11 a a a a , a 11 a 22 and a 12 a 21 ← One way of listing all elementary products is to fix the first index and the second one open for permutation....
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 Spring '07
 Chung
 Linear Algebra, Algebra, Eigenvectors, Vectors, Scalar, Hanyang University, SAS Lab.

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