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Eigenvalues and Eigenvectors Note2.0

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

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Eigenvalues and Eigenvectors: Additional Notes

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1 2 1 6 1 0 1 2 1 A = - - - - 1 2 3 1 1 2 6 , 2 , 3 , 13 1 2 C C C - = = = - 1 2 3 0 4 6 0 , 8 , 9 , 0 4 6 AC AC AC     = = - =     - -   1 1 2 2 3 3 0 , 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 λ =
1 1 2 6 2 3 13 1 2 P - = - - 1 7 0 7 1 27 24 9 84 32 12 8 P - - - = - 1 7 0 7 1 2 1 1 1 2 0 0 0 1 27 24 9 6 1 0 6 2 3 0 4 0 84 32 12 8 1 2 1 13 1 2 0 0 3 P AP - - - -         = - - = -         - - - - -     1 0 0 0 0 4 0 0 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

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1 0 0 0 0 4 0 0 0 3 P AP - = - 1 0 0 0 0 4 0 0 0 3 A P P - = - 1 0 0 0 0 4 0 0 0 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 λ = 0 0 0 (0 is a zero vector) A λ = = 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) λ such that If such a number λ exists, it is called an eigenvalue of A . The vector C is called eigenvector associated to the eigenvalue λ . Remark. The eigenvector C must be non-zero since we have for any number λ .

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1 2 1 6 1 0 1 2 1 A = - - - - 1 1 2 2 3 3 0 , 4 , 3 , AC C AC C AC C = = = 1 2 3 1 1 2 6 , 2 , 3 , 13 1 2 C C C - = = = -
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Eigenvalues and Eigenvectors Note2.0 - Eigenvalues and...

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