1 Nth power of a square matrix and the Binet Formula for Fibonacci sequence Yue Kwok Choy Given A= 4 −12 −12 11 . We begin to investigate how to find A . (1) The story begins in finding the eigenvalue(s) and eigenvector(s) of A . A real number λ is said to be an eigenvalue of a matrix A if there exists a non-zero column vector v such that A=λ orA−λI=0 (a) Eigenvalues A= 4 −12 −12 11 , = x y , A−λI= 4−λ −12 −12 11−λ x y =0 Now, A−λI=0 has non-zero solution, |A−λI|=0 4−λ −12 −12 11−λ =0 4−λ11−λ−144=0 λ −15λ−100=0 λ−20λ+5=0 ∴ λ =20 or λ = −5 , and these are the eigenvalues. (b) Eigenvectors We usually would like to find the unit eigenvector corresponding to each eigenvalue. The process is called normalization. For λ = 20, 4 −12 −12 11 x y =20 x y x +y =1 ⇔ 4x −12y = 20x −12x +11y = 20y x +y =1 4x +3y =0 x +y =1 ∴ x y = 3/5 −4/5 , which is a unit eigenvector. For λ = −5, 4 −12 −12 11 x y =−5 x y x +y =1 ⇔ 4x −12y = −5x −12x +11y = −5y x +y =1 3x −4y =0 x +y =1 ∴ x y = 4/5 3/5 , which is another unit eigenvector.
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Nth power of a square matrix power of...1 Nth power of a square matrix and the Binet Formula for Fibonacci sequence Yue Kwok Choy Given A= 4 −12 −12 11. We begin to investigate
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Nth power of a square matrix and the Binet Formula for Fibonacci sequence
Yue Kwok Choy
Given A = � 4 −12−12 11 �. We begin to investigate how to find A .
(1) The story begins in finding the eigenvalue(s) and eigenvector(s) of A .
A real number λ is said to be an eigenvalue of a matrix A if there exists a non-zero
column vector v such that A = λor�A − λI� = 0 (a) Eigenvalues