2331_Notes_6o1_fill

2331_Notes_6o1_fill - Math 2331 Linear Algebra Section 6.1...

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Math 2331 Linear Algebra Section 6.1 Eigenvalues and Eigenvectors Consider a function . What do the elements in the domain look like? () : n fx \\ n What do the elements in the range look like? So, we can think of any vector x as the input to a function and Ax as the output. What does Ax do to a vector? Example: Let 12 01 A = ⎝⎠ and find Ax for the following vectors 0 1 x ⎛⎞ = ⎜⎟ 1 1 x = 1 3 x = 1 3 x = 1 0 x =
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What is special about the last one? Another Example - 14 23 A ⎛⎞ = ⎜⎟ ⎝⎠ Find Ax when x is- 0 1 x = 1 0 x = 1 1 x = 2 1 x = The fact that the function f(x) = Ax does not change the direction of certain vectors is very significant.
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Definition: Let A be an n x n matrix. A nonzero vector x is called an eigenvector of the matrix A if there exists a scalar λ such that A xx = . The scalar is called an eigenvalue of A. Note that eigenvalues and eigenvectors go together in pairs, an eigenvector is SPECIFIC to an eigenvalue. However, for a given eigenvalue, there are an infintie number of eigenvecors since all scalar multiples of an eigenvector are also eigenvectors for the same eigenvalue. Eigenvectors are stretched, shrunk and / or reversed by the function f(x) = Ax, but they do not move off the line.
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2331_Notes_6o1_fill - Math 2331 Linear Algebra Section 6.1...

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