Powers of a 2×2 matrix in closed form

Here's something I found surprising: the powers of a 2*2 matrix have a fairly simple closed form. Also, the derivation is only one page [1].

LetA be a 2*2 matrix with eigenvalues and . (3Blue1Brown made a nice jingle for finding the eigenvalues of a 2*2 matrix.)

If = then the nth power of A is given by

If then the nth power of A is given by

Example

Let's do an example with

The eigenvalues are 26 and 3. I chose the matrix entries based on today's date, not to have integer eigenvalues, and was surprised that they turned out so simple [2]. (More along those lines here.)

Here's a little Python code to show that the formula above gives the same result as directly computing the cube of A.

 import numpy as np A = np.matrix([[6, 3], [20, 23]]) m = (6 + 23)/2 p = 6*23 - 3*20  = m + (m**2 - p)**0.5  = m - (m**2 - p)**0.5 print(, ) I = np.eye(2) direct = A*A*A formula = **3*(A - *I)/( - ) + **3*(A - *I)/( - ) print(direct) print(formula)

[1] Kenneth S. Williams. The nth Power of a 2*2 Matrix. Mathematics Magazine, Dec., 1992, Vol. 65, No. 5, p. 336.

[2] I wrote a script to find out how often this happens, and it's more often than I would have guessed. There are 31 dates this year that would give integer eigenvalues if arranged as in the example.

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