7.5we - 7.5/7.6: HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT...

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Unformatted text preview: 7.5/7.6: HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS KIAM HEONG KWA A matrix is said to be complex if all its entries are complex scalars. Likewise, a matrix is said to be real if all its entries are real scalars. Except row vectors and column vectors, all matrices are assumed to be real in the sequel. Consider the linear system of first order differential equations x = Ax , where A is a constant square matrix. We claim that if x ( t ) = e rt is a solution of x = Ax , where r is a scalar and is a nonzero vector, then r is an eigenvalue of A and is an eigenvector associated with r . This is so since x ( t ) = r e rt and Ax ( t ) = A e rt . Hence if x ( t ) = e rt is a solution of x = Ax , then A e rt = r e rt . Since e rt is a nonzero scalar, it follows that A = r . Conversely, if r is an eigenvalue of A and if is an eigenvector belongs to r , then it can likewise be shown that x ( t ) = e rt is a solution of x = Ax . We summarize these observations as follows. Lemma 1. Let A be a constant n n matrix. Then x ( t ) = e rt , where r is a scalar and is a nonzero vector, is a solution of (1) x = Ax if and only if r is an eigenvalue of A and is an eigenvector belongs to r . For linear algebraic reasons as well as the uniqueness of a solution with respect to a given initial condition, if x (1) ( t ) , x (2) ( t ) , , x ( n ) ( t ) are n linearly independent solutions of (1) , then each solution of (1) can be expressed as a linear combination (2) x ( t ) = c 1 x (1) ( t ) + c 2 x (2) ( t ) + + c n x ( n ) ( t ) = n X i =1 c i x ( i ) ( t ) of x ( i ) ( t ) , i = 1 , 2 , ,n , where c i , i = 1 , 2 , ,n , are integration constants. Such a collection of n linearly independent solutions is called Date : March 1, 2011. 1 2 KIAM HEONG KWA a fundamental set of solutions , while the representation of any solution in the form of (2) is referred to as the general solution of (1). Recall from section 7.3 that if A has n distinct eigenvalues r i , i = 1 , 2 , ,n , and if ( i ) is an eigenvector belongs to r i for each i , then the eigenvectors ( i ) s are linearly independent. Consequently, (3) x (1) ( t ) = (1) e r 1 t , x (2) ( t ) = (2) e r 2 t , , x ( n ) ( t ) = ( n ) e r n t form a fundamental set of solutions of (1). In fact, by lemma 1, each x ( i ) ( t ) is a solution of (1). On the other hand, let c 1 ,c 2 , ,c n be scalars such that c 1 x (1) ( t ) + c 2 x (2) ( t ) + + c n x ( n ) ( t ) = for all t . Then, in particular, c 1 x (1) (0) + c 2 x (2) (0) + + c n x ( n ) (0) = , from which it follows that c 1 (1) + c 2 (2) + + c n ( n ) = ....
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7.5we - 7.5/7.6: HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT...

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