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].

Let A be a 2×2 matrix with eigenvalues α and β. (3Blue1Brown made a nice jingle for finding the eigenvalues of a 2×2 matrix.)

If α = β then the nth power of A is given by

A^n = alpha^{n-1}left( nA - (n-1)alpha Iright)

If α ≠ β then the nth power of A is given by

A^n = frac{alpha^n}{alpha - beta} (A - beta I) + frac{beta^n}{beta-alpha}(A - alpha I)

Example

Let’s do an example with

A = begin{bmatrix} 6 & 3 \ 20 & 23 end{bmatrix}

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
    α = m + (m**2 - p)**0.5
    β = m - (m**2 - p)**0.5
    print(α, β)
    
    I = np.eye(2)
    direct = A*A*A
    
    formula = α**3*(A - β*I)/(α - β) + β**3*(A - α*I)/(β - α)
    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.

The post Powers of a 2×2 matrix in closed form first appeared on John D. Cook.

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