Download Computing Powers and Square Roots of Diagonalizable Matrices and more Study notes Linear Algebra in PDF only on Docsity!
Lecture 32
Andrei Antonenko
April 30, 2003
1 Powers of diagonalizable matrices
In this section we will give 2 algorithms of computing the m-th power of a matrix.
1.1 Method 1
First method is based on diagonalization. Suppose A is a given matrix, and we want to find its m-th power, i.e. we want to get a formula for Am. We will suppose that the matrix A is diagonalizable. Let λ 1 , λ 2 ,... , λn be the eigenvalues of A, and e 1 , e 2 ,... , en be its linearly independent eigenvectors. Then we know, that there exists a matrix C, whose columns are eigenvectors, and a diagonal matrix
D =
λ 1 0... 0 0 λ 2... 0
............... 0 0... λn
such that D = C−^1 AC, or A = CDC−^1.
Now we can see that
Am^ = (CDC−^1 )m = ( ︸CDC −^1 )(CDC︷︷− 1 ) · · · (CDC−^1 ︸) m = CD(C−^1 C)D(C−^1... C)DC−^1 = CDID... IDC−^1 = CDmC−^1.
But
Dm^ =
λ 1 0... 0 0 λ 2... 0
............... 0 0... λn
m
λm 1 0... 0 0 λm 2... 0
................. 0 0... λmn
So,
Am^ = C
λm 1 0... 0 0 λm 2... 0
................. 0 0... λmn
C−^1.
Example 1.1. Let’s find the formula for
)m
. The characteristic polynomial is
pA(λ) = λ^2 − 3 λ + 2.
So, the eigenvalues are λ 1 = 1, λ 2 = 2. Let’s compute eigenvectors.
λ 1 = 1. After subtraction λ 1 = 1 from the diagonal, we have
, so the eigenvector is (1, 1).
λ 2 = 2. After subtraction λ 2 = 2 from the diagonal, we have
, so the eigenvector is (2, 1).
Thus,
D =
, C =
, C−^1 =
Now we get
Am^ = CDmC−^1
=
)m ( − 1 2 1 − 1
0 2 m
1 2 m+ 1 2 m
−1 + 2m+1^2 − 2 m+ −1 + 2m^2 − 2 m
1.2 Method 2
Let A be an n × n-matrix. Suppose it has n different eigenvalues. Then the algorithm goes as following. Let’s write the approximation equation:
an− 1 tn−^1 + an− 2 tn−^2 + · · · + a 1 = tm
and substituting t = 4, we get 4 a + b = 4m.
Now, taking the derivative of this equation we have
a = mtm−^1 ,
and substituting t = 4, we get a = m 4 m−^1.
So, a = m 4 m−^1 , and b = 4m^ − m 4 m. So, the formula for the m−-th power of A is
Am^ = (m 4 m−^1 )A + (4m^ − m 4 m)I
=
2 m 4 m−^1 + 4m^ − m 4 m^ −m 4 m m 4 m−^1 6 m 4 m−^1 + 4m^ − m 4 m
3 Square roots of diagonalizable matrices
In the previous chapters we saw how to compute m-th power of a diagonalizable matrix using eigenvectors and eigenvalues. Now we will consider a problem of finding a square root of a matrix. Suppose the matrix A is diagonalizable, i.e. it has n linearly independent eigenvectors e 1 , e 2 ,... , en with corresponding eigenvalues λ 1 , λ 2 ,... , λn. Now if C is a matrix, where ei’s are written as columns, and D is a diagonal matrix with λi’s over diagonal, then
A = CDC−^1.
Let all λi’s be nonnegative numbers. Now let’s consider a matrix
D which has either positive or negative square roots of λi’s on diagonal:
D =
λ 1 0... 0 0 ±
λ 2... 0
.......................... 0 0... ±
λn
We can easily check that for such defined
D, we have (
D)^2 = D. So we see that a square root of a matrix A can be obtained from D and C by the following formula:
C
DC−^1.
Let’s denote that there are more than one square root of a matrix — and all of them can be obtained by choosing different signs before
λi’s in D−^1.
Example 3.1. Let’s compute a square root of A =
. The characteristic polynomial is
pA(λ) = λ^2 − 13 λ + 36, so eigenvalues are λ 1 = 4, λ 2 = 9. Eigenvector, corresponding to λ 1 = 4
is determined from equation 3 x 1 + 2x 2 = 0, so it can be (2, −3). Eigenvector, corresponding to λ 2 = 9 is determined from equation − 2 x 1 + 2x 2 = 0, so it can be (1, 1). So,
D =
, C =
, C−^1 =^1
Now, √ D =
So, for positive signs in D we have:
√ A =^15
=^1
In the same way we can get other square roots (changing signs in D).
If A has negative eigenvalues, then this algorithm is not applicable, but in this case there are no square roots of A.
