10 - Eigenvalues and Eigenvector 2

10 - Eigenvalues and Eigenvector 2 - 1 Math 1b Practical...

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1 Math 1b Practical — Eigenvalues and eigenvectors, II February 11, 2011 Example 6. Let P be the matrix of the orthogonal projection onto a subspace U of R n . If u U ,then P u = u ,andif w U P w = 0. That is, elements of U are eigenvec- tors corresponding to eigenvalue 1, and elements of U are eigenvectors corresponding to eigenvalue 0. The (geometric) multiplicity of 1 is dim( U ) and the (geometric) multiplicity of 0 is n dim( U ). Let R be the matrix of the reflection through a subspace U of R n .I f u U R u = u ,andi f w U ,th en R w = w . That is, elements of U are eigenvectors corresponding to eigenvalue +1, and elements of U are eigenvectors corresponding to eigenvalue 1. (I may give a more careful explanation in class.) Example 7. If A and B are square matrices, the eigenvalues of A O O B are those of A AND those of B . More about this in class, or in a later version of this handout. ∗∗∗ In Example 1 of the part I of this handout, we saw that 5 and 2 were the eigenvalues of A = ± 13 42 ² and that A e 1 = ³ ´³ 3 4 ´ = ³ 15 20 ´ =5 e 1 , A e 2 = ³ 1 1 ´ = ³ 2 2 ´ = 2 e 2 . As an indication of why this helps us understand A and its powers, ±rst note that A n e 1 n e 1 and A n e 2 =( 2) n e 2 . Since e 1 and e 2 form a basis for R 2 , we can understand A n x by writing x as a linear combination of e 1 and e 2 . For example, let x =(9 , 5) > .W ehav e( che ckth i s ) ³ 9 5 ´ =2 ³ 3 4 ´ +3 ³ 1 1 ´
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2 and then A n ± 9 5 ² =2 A n ± 3 4 ² +3 A n ± 1 1 ² · 5 n ± 3 4 ² +3 · ( 2) n ± 1 1 ² = ± 6 · 5 n +3 · ( 2) n 8 · 5 n 3 · ( 2) n ² . Thatis,wehavederiveda formula for A n (9 , 5) > . One can see from this, for example, that the ratio of its Frst and second coordinates tends to 3 / 4as n tends to inFnity. Sometimes a matrix has an positive eigenvalue λ so that λ> | μ | for every other eigenvalue μ ; such an eigenvalue is called the dominant eigenvalue . In this example, 5 is the dominant eigenvalue.
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10 - Eigenvalues and Eigenvector 2 - 1 Math 1b Practical...

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