Linear Systems of First-Order Differential Equations 349In exercises 37-40 show that the given set of vector functions are
linearly independent on the interval specified.
8.2 Eigenvalues and Eigenvectors
DEFINITIONS Eigenvalues and Eigenvectors of a Matrix
The scalar A is an eigenvalue (or characteristic value) of the n x n,
constant matrix A and the nonzero n x 1 vector x is a n eigenvector (or
characteristic vector) associated with A if and only if Ax= Ax.Thus, an eigenvector of the matrix A is a nonzero vector which when multi-
plied by A equals some constant, A, times itself. A vector chosen at random
will generally not have this property. However, if x is a vector such that
Ax = Ax, then y = ex also satisfies Ay = Ay for any arbitrary constant c,
since Ay = A(cx) = (Ac)x = (cA)x = c(Ax) = cAx = Acx = Ay. That is,
if x is an eigenvector of the matrix A associated with the eigenvalue A, then
so is ex. Hence, eigenvectors are not uniquely determined but are determined
only up to an arbitrary multiplicative constant. If the manner in which the
arbitrary constant is to be chosen is specified, then the eigenvector is said to
be normalized. For example, the eigenvectors x of an n x n matrix A could
be normalized so that (xi+ x§ + · · · + x~)^1!^2 = 1 where x1, x2, ... , Xn denote
the components of x.
There is a nonzero vector x which satisfies the equation(1) Ax= AX