M 7 3 3 7 Find formulas for the entries of M where n is a p

M = [-7 3 -3 -7] Find formulas for the entries of M\". where n is a positive integer. (Your formulas should not contain complex numbers.)

Solution

It\'s easy to compute powers of diagonal matrices. In particular,
[a 0]^n..[a^n 0]
[0 b]...=[0 b^n]

(forgive the awful typesetting). If you can diagonalize your matrix M, you can put it in the form M = PDP\' (where \' denotes the inverse) where P is invertible and D is diagonal. If you take this equation to the 2nd power, you get M^2 = (PDP\')(PDP\') = (PD)(P\'P)(DP\') = (PD)I(DP\') = P(DID)P\' = P(DD)P\' = P D^2 P\'. Similarly, if you take it to the nth power, you get M^n = P D^n P\'.

So, if you can diagonalize M, you\'re basically there. To diagonalize a matrix, you can compute the eigenvalues and eigenvectors, which (so long as there are \"enough\" of them) can be placed directly in D and P.

Eigenvalue/vector computations:
(11 - t) * (2 - t) + 6*3 = 0
=> t=5 or 8

For t=5:
M - 5I =
[6 -6]
[3 -3]

by inspection has (1, 1) sent to 0, so that (1, 1) is an eigenvector of M corresponding to eigenvalue 5.

For t=8:
M - 8I =
[3 -6]
[3 -6]

by inspection has (2, 1) sent to 0, so that (2, 1) is an eigenvector of M corresponding to eigenvalue 8.


D\'s entries will be the eigenvalues and P\'s columns will be the corresponding eigenvectors (in the same order), so we have

D =
[5 0]
[0 8]

P =
[1 2]
[1 1]

(So P\' =
[-1 2]
[1 -1])


Now we have
M^n =
[1 2][5^n 0][-1 2]
[1 1][0 8^n][1 -1]
=
[5^n 2*8^n][-1 2]
[5^n 8^n]...[1 -1]
=
[2*8^n - 5^n ... 2*5^n - 2*8^n]
[8^n - 5^n ... 2*5^n - 8^n]