chapter3.page09 - 1. VECTORS AND MATRICES 9 - The above...

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Unformatted text preview: 1. VECTORS AND MATRICES 9 - The above definition of eigenvector and eigenvalue is valid for any square matrix with n rows and columns. - p() = |A-I| is a polynomial of degree n for nn matrix A, which is called the characteristic polynomial of A. - If we view A as an transform that maps a vector x to Ax, an eigenvector v defines a straight line passing origin that is invariant under A. - If v is an eigenvector then for and number s = 0, sv is also an eigenvector. This is especially useful when using Mathcad to get eigenvectors, the result of Mathcad might look "bad", you might need to remove the common factor of the component of the vector to make it "better." Computing eigenvalues and eigenvectors of a given matrix is quite tedious, Mathcad provides two functions eigenvals() and eigenvecs() to compute eigenvalues and eigenvectors of a matrix. In Mathcad , eigenvecs(M) Returns a matrix containing the eigenvectors. The nth column of the matrix returned is an eigenvector corresponding to the nth eigenvalue returned by eigenvals. The results of these functions by default is in decimal, you can change it by using simplify key word as shown in the following diagram. (a) Find eigenvalue (b) Find eigenvector Figure 3. Compute eigenvalue and eigenvector in Mathcad 3 - 3 Notice, in the diagram, the eigenvalues are listed as vector and the eigenvectors are listed in a matrix -( 3-1) 1+ 3 1 1 (8+2 3) 2 (8-2 3) 2 , 2 2 1 1 (8+2 3) 2 (8-2 3) 2 ...
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This note was uploaded on 10/04/2011 for the course MAP 4231 taught by Professor Dr.han during the Spring '11 term at UNF.

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