Nonlinear Control Sy.. - Free

2.4. MATRICES 39
2.4 Matrices
We assume that the reader has some acquaintance with the elementary theory of matrices
and matrix operations. We now introduce some notation and terminology as well as some
useful properties.
Transpose: If A is an m x n matrix, its transpose, denoted AT, is the n x m matrix obtained
by interchanging rows and columns with A. The following properties are straightforward
to prove:
(AT)T = A.
(AB)T = BT AT (transpose of the product of two matrices)-
(A + B)T =A T +B T (transpose of the sum of two matrices).
Symmetric matrix: A is symmetric if A = AT.
Skew symmetric matrix: A is skew symmetric if A = -AT.
Orthogonal matrix: A matrix Q is orthogonal if QTQ = QQT = I. or equivalently, if
QT
Inverse matrix: A matrix A-1 E R"' is said to be the inverse of the square matrix
A E R""" if
AA-1 = A-1A = I
It can be verified that
(A-1)-' = A.
=Q-1
(AB)-1 = B-'A-1, provided that A and B are square of the same size, and invertible.
Rank of a matrix: The rank of a matrix A, denoted rank(A), is the maximum number
of linearly independent columns in A.
Every matrix A E 1R""" can be considered as a linear function A : R' -4 IR", that is
the mapping
y = Ax
maps the vector x E ]R" into the vector y E R'.