Eigenvalues and Eigenvectors Note2.0

Eigenvalues and Eigenvectors Note2.0 - Eigenvalues and...

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Unformatted text preview: Eigenvalues and Eigenvectors: Additional Notes 1 2 1 6 1 1 2 1 A =- --- 1 2 3 1 1 2 6 , 2 , 3 , 13 1 2 C C C- = = = - 1 2 3 4 6 0 , 8 , 9 , 4 6 AC AC AC = = - = -- 1 1 2 2 3 3 , 4 , 3 , AC C AC C AC C = = = Example. Consider the matrix Consider the three column matrices We have In other words, we have 0,4 and 3 are eigenvalues of A, C 1 ,C 2 and C 3 are eigenvectors Ac c = 1 1 2 6 2 3 13 1 2 P- = -- 1 7 7 1 27 24 9 84 32 12 8 P--- =- 1 7 7 1 2 1 1 1 2 1 27 24 9 6 1 6 2 3 4 84 32 12 8 1 2 1 13 1 2 3 P AP---- =-- =- ----- 1 4 0 3 P AP- =- Consider the matrix P for which the columns are C 1 , C 2 , and C 3 , i.e., we have det ( P ) = 84. So this matrix is invertible. Easy calculations give Next we evaluate the matrix P-1 AP . In other words, we have 1 4 3 P AP- =- 1 4 3 A P P- =- 1 4 3 for 1,2,... n n n A P P n- =- = In other words, we have Using the matrix multiplication, we obtain which implies that A is similar to a diagonal matrix. In particular, we have . AC C = (0 is a zero vector) A = = Definition. Let A be a square matrix. A non-zero vector C is called an eigenvector of A if and only if there exists a number (real or complex) such that If such a number exists, it is called an eigenvalue of A . The vector C is called eigenvector associated to the eigenvalue ....
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This note was uploaded on 09/12/2011 for the course E 101 taught by Professor J during the Spring '11 term at Adrian College.

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Eigenvalues and Eigenvectors Note2.0 - Eigenvalues and...

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