{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

m2k_dfq_cstcof

m2k_dfq_cstcof - Constant Coecient Linear Systems Denition...

This preview shows pages 1–4. Sign up to view the full content.

Constant Coefficient Linear Systems Definition A constant coefficient system of linear differential equa- tions takes the form Y prime ( t ) = A Y ( t ) + B ( t ) , where A is a constant n × n matrix. As usual, the system is homogeneous if B ( t ) 0, inhomogenous otherwise. Example 1 The system y prime 1 ( t ) y prime 2 ( t ) y prime 3 ( t ) = 2 1 0 1 3 1 0 1 2 y 1 ( t ) y 2 ( t ) y 3 ( t ) is a constant coefficient, first order, three dimensional, homogeneous system. When we have a constant coefficient scalar system z prime ( t ) = a z ( t ) the general solution takes the form z ( t, c ) = c e at , where c is an arbi- trary constant. Since the solutions of a vector system must be vector functions, it makes sense to try for a solution of the form Y ( t, λ, Φ) = Φ e λt , Φ R n , where λ is a constant (scalar) and Φ is a constant vector. Substituting this solution form into the system Y prime ( t ) = A Y ( t ) we have λ Φ e λt = A Φ e λt -→ ( λ I n - A ) Φ e λt = 0 . Since the exponential cannot vanish, we have ( λ I n - A ) Φ = 0 , i . e ., A Φ = λ Φ; 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
thus A Φ is just a scalar multiple of Φ , λ being the scalar. Definition If λ is a number such that, for some non-zero n dimen- sional vector, Φ, ( λ I n - A ) Φ = 0 then λ is said to be an eigenvalue of the matrix A and Φ is an eigen- vector of A corresponding to λ . Proposition There exists a non-zero vector Φ such that ( λ I n - A ) Φ = 0 if and only if λ is such that det ( λ I n - A ) p ( λ ) = 0; here p ( λ ) is an n -th degree polynomial, p ( λ ) = λ n + a 1 λ n - 1 + · · · + a n - 1 λ + a n , which we call the characteristic polynomial of the matrix A . The fact that the determinant must be equal to zero for Φ negationslash = 0 to exist as shown is a familiar result from linear algebra. If we let the columns of A be A j and let the columns of the identity matrix be E j , j = 1 , 2 , ..., n, then p ( λ ) = det ( λ I n - A ) = det [ λ E 1 - A 1 , λ E 2 - A 2 , · · · , λ E n - A n ] = λ n det I n + · · · + ( - 1) n det A , using the linearity of the determinant with respect to the columns of the matrix in question and the form of p ( λ ) then follows. Example 2 For the 3 × 3 matrix in the system shown at the beginning of the lecture we have det ( λ I n - A ) = det λ - 2 - 1 0 - 1 λ - 3 - 1 0 - 1 λ - 2 = ( λ - 2)[( λ - 3)( λ - 2) - 1] - ( λ - 2) = λ 3 - 7 λ 2 + 14 λ - 8 2
= ( λ - 1)( λ - 2)( λ - 4); the eigenvalues are 1 , 2 , and 4. It should be noted that eigenvectors are not quite unique; if Φ is an eigenvector of A corresponding to the eigenvalue λ and if c negationslash = 0 is a constant, then c Φ is also an eigenvector because it is a non-zero vector for which ( λ I n - A )( c Φ) = 0. Very commonly, in order to fix a particular eigenvector one chooses c = 1 bardbl Φ bardbl , in which case we obtain the eigenvector for which bardbl c Φ bardbl = vextenddouble vextenddouble vextenddouble vextenddouble 1 bardbl Φ bardbl Φ vextenddouble vextenddouble vextenddouble vextenddouble = 1 bardbl Φ bardbl bardbl Φ bardbl = 1. This process is called normalization and the resulting eigenvector is the normalized eigenvector.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}