Nonlinear Control Sy.. - Free

42 CHAPTER 2. MATHEMATICAL PRELIMINARIES
any matrix A E 1R"' can be rewritten as the sum of a symmetric and a skew symmetric
matrix, as shown below:
A= 2(A+AT)+2(A-AT)
Clearly,
B = 2(A+AT) = BT, and
C = 2(A-AT)=-CT
thus B is symmetric, whereas C is skew symmetric. For the skew symmetric part, we have
that
xTCx = (xTCx)T = xTCTx = -xTCx.
Hence, the real number xT Cx must be identically zero. This means that the quadratic form
associated with a skew symmetric matrix is identically zero.
Definition 2.11 Let A E ]R""" be a symmetric matrix and let x
be:
(i) Positive definite if xTAx > OVx # 0.
(ii) Positive semidefinite if xT Ax > OVx # 0.
(iii) Negative definite if xT Ax < OVx # 0.
(iv) Negative semidefinite if xT Ax < OVx # 0.
(v) Indefinite if xT Ax can take both positive and negative values.
EIIF".
Then A is said to
It is immediate that the positive/negative character of a symmetric matrix is determined
completely by its eigenvalues. Indeed, given a A = AT, there exist P such that P-'AP =
PT AP = D. Thus defining x = Py, we have that
yTPTAPy
yT P-1APy
yTDy
'\l yl + A2 y2 + ... + .1"y2n
where A,, i = 1, 2, n are the eigenvalues of A. From this construction, it follows that