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EIGENVALUES, EIGENVECTORS Pertemuan 6 Matakuliah: MATRIX ALGEBRA FOR STATISTICS Tahun: 2009

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A is an m x m matrix, then any scalar satisfying the equation AX = X, for some mx1 vector X 0, is called an eigenvalue of A. The vector X is called an eigenvector of A corresponding to the eigenvalue A, and equation AX = X is called the eigenvalue-eigenvector equation of A. Bina Nusantara University

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4 Eigenvalues and eigenvectors are also sometimes referred to as latent roots and vectors or characteristic roots and vectors. AX = X dapat diubah menjadi (A - I)X = 0

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Eigenvalue A must satisfy (A - I) = 0, called characteristic equation of A There are scalars α 0,..., α m-1 such that the characteristic equation above can be expressed α 0 +α 1 (- )+... +α m-1 (- ) m-1 +(- ) m Bina Nusantara University 5

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Since an m th degree polynomial has m roots, it follows that an mxm matrix has m eigenvalues; that is, there are m scalars 1,..., m, which satisfy the characteristic equation Bina Nusantara University 6

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Contoh: Bina Nusantara University 7 Find the eigenvalues and eigenvectors of matrix A The characteristic equation of A is

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= -(5 - ) 2 (2 + ) - 3(4) 2 - 4(3) 2 + 3(4)(2 + ) + 3(4)(5 - ) + 3(4)(5 - ) = - 3 + 8 2 - 17 + 10 = -( - 5)( - 2)( - 1) = 0 The three eigenvalues of A are 1, 2, and 5 Bina Nusantara University 8

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To find the eigenvector For =1, solve the equation Ax = 1x for x, which yields the system of equations 5x 1 - 3X 2 + 3X 3 = x 1 4x 1 - 2x 2 + 3X 3 = X 2 4x 1 - 4X 2 + 5X 3 = X 3 The eigenvector for eigenvalue 1 is Find the eigenvector for eigenvalue 2 and 5! Bina Nusantara University 9

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Application: Covariance matrix Multivariate analyses (lihat buku 2 hal. 100- 120 ) Bina Nusantara University 10

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Computing Eigen Information for Small Matrices The eigen equation can be rearranged as follows: Ax = x Ax = I n x Ax - I n x = 0 (A - I n )x = 0.

Computing Eigen Information for Small Matrices The eigen equation can be rearranged as follows: Ax = x Ax = I n x Ax - I n x = 0 (A - I n )x = 0.

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