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Powers of a 2×2 matrix in closed type

Powers of a 2×2 matrix in closed type

2023-06-11 05:39:20

Right here’s one thing I discovered stunning: the powers of a 2×2 matrix have a reasonably easy closed type. Additionally, the derivation is just one web page [1].

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

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

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

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

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

Instance

Let’s do an instance with

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

The eigenvalues are 26 and three. I selected the matrix entries primarily based on at the moment’s date, to not have integer eigenvalues, and was stunned that they turned out so easy [2]. (Extra alongside these strains here.)

See Also

Right here’s a little bit Python code to indicate that the components above provides the identical consequence as immediately computing the dice 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
    
    components = α**3*(A - β*I)/(α - β) + β**3*(A - α*I)/(β - α)
    print(direct)
    print(components)
[1] Kenneth S. Williams. The nth Energy of a 2×2 Matrix. Arithmetic Journal, Dec., 1992, Vol. 65, No. 5, p. 336.

[2] I wrote a script to learn the way typically this occurs, and it’s extra typically than I might have guessed. There are 31 dates this yr that might give integer eigenvalues if organized as within the instance.

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