chapter4

chapter4 - Math 152: Linear Systems Winter 2005 Section 4:...

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Unformatted text preview: Math 152: Linear Systems Winter 2005 Section 4: Eigenvalues and Eigenvectors Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Computing the eigenvalues and eigenvectors (Examples 1 . . . 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Complex numbers (Example 5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Eigenvalues and eigenvectors: summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Complex exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Polar representation of a complex number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Systems of linear differential equations (Examples 1 . . . 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Computing high powers of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Another formula for the determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 The matrix exponential and differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 LCR circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Converting higher order equations into first order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Springs and weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 c 2003 Richard Froese. Permission is granted to make and distribute verbatim copies of this document provided the copyright notice and this permission notice are preserved on all copies. 84 Math 152 Winter 2003 Section 4: Eigenvalues and Eigenvectors Eigenvalues and eigenvectors Let A be an n n matrix. A number and a vector x are called an eigenvalue eigenvector pair if (1) x 6 = (2) A x = x In other words, the action of A on the vector x is to stretch or shrink it by an amount without changing its direction.its direction....
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chapter4 - Math 152: Linear Systems Winter 2005 Section 4:...

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