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2. Functions of matrix

In a discrete dynamical system, we compute the power of linear transformation (or matrix). From the matrix viewpoint, what we have done is the following. Suppose A = PDP-1 for an invertible P and diagonal

D = [ λ1 0 . . 0 ].
0 λ2 . . 0
: :   :
0 0 . . λn

The power of the diagonal matrix is easy to compute (try the case n = k = 3 to convince yourself)

Dk = [ λ1k 0 . . 0 ].
0 λ2k . . 0
: :   :
0 0 . . λnk

Thus by P-1P = I we have

Ak = (PDP-1)k = PDP-1PDP-1...PDP-1 = PDkP-1 = P[ λ1k 0 . . 0 ]P-1.
0 λ2k . . 0
: :   :
0 0 . . λnk

We see it is quite easy to compute the powers of a matrix from its diagonalization.

Example From an earlier example, we have

[ 13 -4 ] = [ 1 2 ] [ 5 0 ] [ 1 2 ]-1.
-4 7 -2 1 0 15 -2 1

Thus

[ 13 -4 ]k = [ 1 2 ] [ 5k 0 ] [ 1 2 ]-1
-4 7 -2 1 0 15k -2 1
= (1/5)[ 5k + 4•15k -2•5k + 2•15k ] = 5k-1[ 1 + 4•3k -2 + 2•3k ].
-2•5k + 2•15k 4•5k + 15k -2 + 2•3k 4 + 3k

The computation of the power of a diagonalizable matrix A = PDP-1 can be extended to other functions. For the polynomial p(t) = 1 + 2t + 3t2, we have

p(A) = I + 2A + 3A2 = I + 2PDP-1 + 3(PDP-1)2 = P(I + 2D + 3D2)P-1

= P[ 1 + 2λ1 + 3λ12 0 . . 0 ]P-1.
0 1 + 2λ2 + 3λ22 . . 0
: :   :
0 0 . . 1 + 2λn + 3λn2

For the exponential function

et = 1 + t + 1 t2 + 1 t3 + ... + 1 tk + ... ,
2! 3! k!

we find

eD = I + D + 1 D2 + 1 D3 + ... + 1 Dk + ...
2! 3! k!

to be a diagonal matrix with the exponential of the eigenvalues

eλ = 1 + λ + 1 λ2 + 1 λ3 + ... + 1 λk + ...
2! 3! k!

as the diagonal entries. Then we have

eA = P(I + D + 1 D2 + 1 D3 + ... + 1 Dk + ...)P-1 = PeDP-1.
2! 3! k!

In general, for a function f(t) and a diagonalizable matrix A = PDP-1, we would expect

f(A) = P[ f(λ1) 0 . . 0 ]P-1.
0 f(λ2) . . 0
: :   :
0 0 . . f(λn)

Strictly speaking, for the similar argument to work, we need f(t) to have power series expansion at t = 0 and have to worry about the convergence issue. The existence of power series means that the function f(t) should be analytic at 0. The convergence means that the "size" of A (called the norm) should be less than the radius of convergence for the power series.

By a more advanced theory, it is possible to define the continuous functions of symmetric matrices (or self-adjoint linear transformations).


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