where V is a
nonsingular matrix and C, N are square Jordan blocks corresponding to eigenvalues lying in [C.sup.-] (open left-half complex plane) and [[??].sup.+] (open right-half complex plane), respectively.
Since B is a
nonsingular matrix based on the algebraic knowledge one can have [x.sup.*] - [y.sup.*] = 0 or [x.sup.*] = [y.sup.*].
if there exists a
nonsingular matrix S (t) with real entries such that
Thus, we can find a
nonsingular matrix P consisting of the n independent eigenvectors of A so that
Let us apply a transformation of variable x(k) = Q[??](k) with a
nonsingular matrix Q to system (3):
for all t [member of] R, any k [member of] {0, 1, ..., n} and any i [member of] {1, ..., n}, where N is a
nonsingular matrix such that [N.sup.-1] AN is a Jordan form matrix and [[[N.sup.-1][[vector].[upsilon]](s)].sub.i] denotes the i-th component of [N.sup.-1][[vector].[upsilon]](s).
It is well known that a
nonsingular matrix over any field has a unique inverse.
Let S be the
nonsingular matrix that contains these eigenvectors as columns [S.sub.:k], and let [Lambda] = Diag([[Lambda].sub.k]) be the associated diagonal matrix of eigenvalues [[Lambda].sub.k] (k = 1, ..., m).
Let a
nonsingular matrix A [member of] [C.sup.n x n] and a vector b [member of] [C.sup.n] be given.
with the
nonsingular matrix (<[L.sub.k], [[??].sup.j]>), k, j = [bar.1,n], since (2.5) is valid for every fundamental system and, particulary, for [z.sup.j] = [[??].sup.j], j = [bar.1,n].
It is known that, for every matrix A, there exists a
nonsingular matrix S transforming it to the corresponding Jordan matrix form [LAMBDA] = [S.sup.-1] AS.