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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #10 (due Thursday, November 17) All problems are from Leons book (8th edition). Section 5.5: 6c, 29a, 29b, 30a, 30b Section 5.6: 1a, 3, 4, 7, 8 MATH 304 Linear Algebra Lecture 22: Eigenvalues and eigenvectors (continued). Characteristic polynomial. Eigenvalues and eigenvectors of a matrix Definition. Let A be an n n matrix. A number R is called an eigenvalue of the matrix A if A v = v for a nonzero column vector v R n . The vector v is called an eigenvector of A belonging to (or associated with) the eigenvalue . Remarks. Alternative notation: eigenvalue = characteristic value , eigenvector = characteristic vector . The zero vector is never considered an eigenvector. Diagonal matrices Let A be an n n matrix. Then A is diagonal if and only if vectors e 1 , e 2 , . . . , e n of the standard basis for R n are eigenvectors of A . If this is the case, then the diagonal entries of the matrix A are the corresponding eigenvalues: A = 1 O 2 . . . O n A e i = i e i Eigenspaces Let A be an n n matrix. Let v be an eigenvector of A belonging to an eigenvalue . Then A v = v = A v = ( I ) v = ( A I ) v = . Hence v N ( A I ), the nullspace of the matrix A I . Conversely, if x N ( A I ) then A x = x . Thus the eigenvectors of A belonging to the eigenvalue are nonzero vectors from N ( A I ). Definition. If N ( A I ) negationslash = { } then it is called the eigenspace of the matrix A corresponding to the eigenvalue . How to find eigenvalues and eigenvectors? Theorem Given a square matrix A and a scalar , the following statements are equivalent: is an eigenvalue of A , N ( A I ) negationslash = { } , the matrix A I is singular, det( A I ) = 0. Definition. det( A I ) = 0 is called the characteristic equation of the matrix A . Eigenvalues of A are roots of the characteristic equation. Associated eigenvectors of A are nonzero solutions of the equation ( A I ) x = . Example. A = parenleftbigg a b c d parenrightbigg . det( A I ) = vextendsingle vextendsingle vextendsingle vextendsingle a b c d vextendsingle vextendsingle vextendsingle vextendsingle = ( a )( d ) bc = 2 ( a + d ) + ( ad bc ). Example. A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ....
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This note was uploaded on 03/19/2012 for the course STAT 211 taught by Professor Parzen during the Spring '07 term at Texas A&M.
 Spring '07
 Parzen

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