CONTINUUM MECHANICS for ENGINEERS

Addition of matrices is commutative, A + B = B + A, and associative,
A +(B + C) = (A + B) + C.
Example 2.4-1
Show that the square matrix A can be expressed as the sum of a symmetric
and a skew-symmetric matrix by the decomposition
A + A A − A
A = +
2 2
T T
Solution
Let the decomposition be written as A = B+ C where B A A and
. Then writing B and C in element form,
= 1
( + ) 2
T
1
T
C= ( A – A ) 2
B
C
ij
ij
A + A A + A A + A
= = =
2 2 2
T T
ij ij ij ji ji ji
A − A A − A A − A
= = =−
2 2 2
T T
ij ij ij ji ji ji
= B = B
T
ji ij
=− C =−C
T
ji ij
Therefore, B is symmetric, and C skew-symmetric.
(symmetric)
(skew-symmetric)
Multiplication of the matrix A by the scalar λ results in the matrix λA, or
[λA ij]. The product of two matrices A and B, denoted by AB, is defined only
if the matrices are conformable, that is, if the prefactor matrix A has the same
number of columns as the postfactor matrix B has rows. Thus, the product
of an M × Q matrix multiplied by a Q × N matrix is an M × N matrix. The
product matrix C = AB has elements given by
C = A B
ij ik kj
(2.4-3)
in which k is, of course, a summed index. Therefore, each element C ij of the
product matrix is an inner product of the i th row of the prefactor matrix with
the j th column of the postfactor matrix. In general, matrix multiplication is
not commutative, AB ≠ BA, but the associative and distributive laws of
multiplication do hold for matrices. The product of a matrix with itself is
the square of the matrix, and is written AA = A 2 . Likewise, the cube of the
matrix is AAA = A 3 , and in general, matrix products obey the exponent rule
A A = A A = A
m n n m m+ n
(2.4-4)