NB. Responding to a request from Raul in Jbeta
NB. A cool use of eigen values and vectors -- recursion
NB. Preliminaries
clean =: * *@| NB. numbers with 2^_44 > | are replaced by 0
mp =: +/ . * NB. matrix product
NB. below, square matrix x to integer power y
mpwr =: ( 4 : 'x mp^:y =i.#x' )"2 0
NB. The recursion ( (1+k) { u ) -: A mp k { u where u is a matrix
]A =: 1 1,.1 0 NB. used for Fibonnaci recursion
]lambda =: 2 %~ (1+%:5),1-%:5 NB. eigenvalues of A
]Lambda =: lambda * =i.# lambda NB. eigenvalues on diagonal
]S =: 2 2 $ lambda,1 1 NB. columns are eigenvectors
(A mp S) -: lambda *&.|: S NB. test claimed eigen vectors, values
]u0 =: 1 0 NB. first row of matrix u
]nr =: 7 NB. number of rows in u
]u =: (A mpwr i. nr) mp"2 1 u0 NB. produce matrix u
NB. Rows obey Fibonnaci recursion, observe second column
(}. u) -: }: A mp&.|: u NB. test "A" recursion in the rows of u
NB. You can produce u using the eigen vectors and values of A:
u -: ( S mp"2 (Lambda mpwr i. nr) mp"2 S mpwr _1 ) mp"2 1 u0
NB. More simply:
c =: (S mpwr _1) mp u0
u -: S mp"2 1 c *"1 lambda^"1 0 i. nr
NB. Above will work if n by n A has n linearly independent eigenvectors.
--Kip Murray
Sent from my iPad
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