Page 58 - Dynamics of Mechanical Systems

0593_C02_fm Page 39 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 39

matrices and on row and column arrays. Recall that a matrix A is simply an array of
numbers a (i = 1,…, m ; j = 1,…, n) arranged in m rows and n columns as:
i
ij
a ... a n
a  11 12 1
 ... 
2
A =  a 21 a 22 a n  (2.10.1)
 ... 
 
 a m1 a 2 n ... a mn
The entries a are usually called the elements of the matrix. The first subscript indicates
ij
the row position, and the second subscript designates the column position.
Two matrices A and B are said to be equal if they have equal elements. That is,

A = if and only if a = b i = ,..., n j = ,..., m (2.10.2)
1
B
1
;
ij
ij
If all the elements of matrix are zero, it is called a zero matrix. If a matrix has only one
row, it is called a row matrix or row array. If a matrix has only one column, it is called a
column matrix or column array. A matrix with an equal number of rows and columns is a
square matrix. If all the elements of a square matrix are zero except for the diagonal
elements, the matrix is called a diagonal matrix. If all the diagonal elements of a diagonal
matrix have the value 1, the matrix is called an identity matrix. If a square matrix has a
T
zero determinant, it is said to be a singular matrix. The transpose of a matrix A (written A )
is the matrix obtained by interchanging the rows and columns of A. If a matrix and its
transpose are equal, the matrix is said to be symmetric. If a matrix is equal to the negative
of its transpose, it is said to be antisymmetric.
Recall that the algebra of matrices is based upon a few simple rules: First, the multipli-
cation of a matrix A by a scalar s produces a matrix B whose elements are equal to the
elements of A multiplied by s. That is,

B = sA if and only if b = sa (2.10.3)
ij ij

Next, the sum of two matrices A and B is a matrix C whose elements are equal to the
sum of the respective elements of A and B. That is,

+
C = A B if andonlyif c = a + b (2.10.4)
ij ij ij
Matrix subtraction is defined similarly.
The product of matrices is defined through the “row–column” product algorithm. The
product of two matrices A and B (written AB) is possible only if the number of columns
in the first matrix A is equal to the number of rows of the second matrix B. When this
occurs, the matrices are said to be conformable. If C is the product AB, then the element c ij
is the sum of products of the elements of the ith row of A with the corresponding elements
of the jth column of B. Specifically,

ij ∑
c = n a b (2.10.5)
ik kj
k=1
where n is the number of columns of A and the number of rows of B.