Systems of Differential Equations - University of Utah
Systems of Differential Equations
Matrix Methods
? Characteristic Equation ? Cayley-Hamilton
? Cayley-Hamilton Theorem ? An Example
? The Cayley-Hamilton-Ziebur Method for u = Au ? A Working Rule for Solving u = Au ? Solving 2 ? 2 u = Au
? Finding d1 and d2 ? A Matrix Method for Finding d1 and d2 ? Other Representations of the Solution u ? Another General Solution of u = Au
? Change of Basis Equation
Characteristic Equation
Definition 1 (Characteristic Equation)
Given a square matrix A, the characteristic equation of A is the polynomial equation
det(A - rI) = 0.
The determinant det(A - rI) is formed by subtracting r from the diagonal of A. The polynomial p(r) = det(A - rI) is called the characteristic polynomial.
? If A is 2 ? 2, then p(r) is a quadratic. ? If A is 3 ? 3, then p(r) is a cubic. ? The determinant is expanded by the cofactor rule, in order to preserve factorizations.
Characteristic Equation Examples
Create det(A - rI) by subtracting r from the diagonal of A.
Evaluate by the cofactor rule.
A=
23 04
,
p(r) =
2-r 3 0 4-r
= (2 - r)(4 - r)
2 3 4
2-r 3 4
A = 0 5 6 , p(r) = 0 5 - r 6 = (2-r)(5-r)(7-r)
007
0 0 7-r
Cayley-Hamilton
Theorem 1 (Cayley-Hamilton)
A square matrix A satisfies its own characteristic equation. If p(r) = (-r)n + an-1(-r)n-1 + ? ? ? a0, then the result is the equation
(-A)n + an-1(-A)n-1 + ? ? ? + a1(-A) + a0I = 0, where I is the n ? n identity matrix and 0 is the n ? n zero matrix.
Cayley-Hamilton Example
Assume
2 3 4 A=0 5 6
007
Then
2-r 3 4
p(r) = 0 5 - r 6 = (2 - r)(5 - r)(7 - r) 0 0 7-r
and the Cayley-Hamilton Theorem says that
0 0 0 (2I - A)(5I - A)(7I - A) = 0 0 0 .
000
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