4.1 - Section 4.1 Eigenvalues and Eigenvectors Definition:...

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Section 4.1 Eigenvalues and Eigenvectors Definition: Let A be an n × n matrix, u be a nonzero n × 1 vector, and λ be a constant. If Au = λ u then is called an eigenvalue for the matrix A and u is called the eigenvector corresponding to .
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Example: Let A = 35 1 1 and u = 10 2 Find the eigenvalue, λ , corresponding to the eigenvector, u . Solution: Au = λ u so 1 1 10 2 = λ 10 2 40 8 = λ 10 2 λ= 4
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Theorem: λ is an eigenvalue for A if and only if A − λ I () is not invertible. Recall: A matrix is not invertible if and only if its determinant is 0. So det A − λ I = 0 when is an eigenvalue and is an eigenvalue when det A − λ I = 0. This will help us find eigenvalues.
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Example: Let A = 5 1 22 Find the eigenvalues, λ . Solution: A − λ I = 5 1 − λ 10 01 = 5 1 0 0 = 5 − λ 1 − λ det( A − λ I ) = 5 − λ 1 − λ = 0
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4.1 - Section 4.1 Eigenvalues and Eigenvectors Definition:...

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