Nonlinear Control Sy.. - Free

2.4. MATRICES 41
Proof: By definition, the columns of the matrix S are the eigenvectors of A. Thus, we have
AS=A :l vn
and we can re write the last matrix in the following form:
Alvl ... Anon
Al
Alvl ... Anvn = V1 ... vn ...
Therefore, we have that AS = SD. The columns of the matrix S are, by assumption,
linearly independent. It follows that S is invertible and we can write
or also
A = SDS-l
D = S-IAS.
This completes the proof of the theorem. El
Special Case: Symmetric Matrices
Symmetric matrices have several important properties. Here we mention two of them,
without proof:
(i) The eigenvalues of a symmetric matrix A E IIF"' are all real.
(ii) Every symmetric matrix A is diagonalizable. Moreover, if A is symmetric, then the
diagonalizing matrix S can be chosen to be an orthogonal matrix P; that is, if A = AT,
then there exist a matrix P satisfying PT = P-l such that
2.4.2 Quadratic Forms
P-'AP =PTAP=D.
Given a matrix A E R""n, a function q : lR - IR of the form
q(x) = xT Ax, x E R"
is called a quadratic form. The matrix A in this definition can be any real matrix. There is,
however, no loss of generality in restricting this matrix to be symmetric. To see this, notice
An