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MatrixAlgebra - Math 17B Kouba Using Eigenvalues and...

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Unformatted text preview: Math 17B Kouba Using Eigenvalues and Eigenvectors to Solve Matrix Algebra Problems DERIVE A FORMULA FOR POWERS OF A MATRIX APPLIED TO ITS EIGENVEC- TOR : Let A be a matrix and assume that A is an eigenvalue with eigenvector V . Then AV 2 AV —> A2V = A(AV) = A(AV) = ,\(AV) 2 /\()\V) = AZV _> A3V = A(A2V) = A(A2V) :2 A2(AV) = A2()\V) = A3V ———> A4V : A(A3V) = A()\3V) = A3(AV) : A3()\V) : A4V ——+ (P) AkV—zxkv for k:1,2,3,4,--- USE THE ABOVE FORMULA TO SOLVE THE FOLLOWING PROBLEM : EXAMPLE : Consider matrix A = (—11/2 3721) . It can be shown that it’s eigenvalues are /\1 = 2 and A2 = —1 with eigenvectcors V1 : (i) and V2 2 (35 ), resp. Use the eigenvalues and eigenvectors for matrix A to determine A30 (123) . Begin by writing vector (123) as a linear combination of the eigenvectors <1) and <25): 2 <23>=<m><a>+<ca><25> - 13: C1 :— 562 2:201+202 —> ~26 : —~201 + 1002 2=2cl+202 —> —24:12C2 —> 022—2 and c1=3. Thus,wehave (123)2(3)(;)+(—2)(‘25) _., 1 ,221,225,482 ,442,450,940 ' ...
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