Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 333

2 3

A 11 þ B 11 A 12 þ B 12 ::: A 1n þ B 1n

4 5 :

A 21 þ B 21 A 22 þ B 22 ::: A 2n þ B 2n

A þ B ¼ (A.10)

: : : : : :

A n1 þ B n1 : : : : A nn þ B nn

Matrix addition is commutative and associative,

A þ B ¼ B þ A and A þð B þ C Þ¼ð A þ B Þþ C ;

(A.11)

respectively. The following distributive laws connect matrix addition and matrix
multiplication by scalars:

að A þ B Þ¼a A þ a B and ða þ bÞ A ¼ a A þ b A ; (A.12)

where are scalars. Negative square matrices may be created by employing

the definition of matrix multiplication by a scalar (A.8) in the special case when
a ¼

a and b

1. In this case the definition of addition of square matrices (A.10) can be
extended to include the subtraction of square matrices, A B .

B T is said to be a symmetric matrix and a matrix for A matrix for which B ¼

C T is said to be a skew-symmetric or anti-symmetric matrix. The
symmetric and anti-symmetric parts of a matrix, say A S and A A , are constructed
from A as follows:

which C ¼

1
2 ð

symmetric part of A S A T ¼ A þ Þ; and (A.13)

1
2 ð

symmetric part of A A A T anti ¼ A Þ: (A.14)

It is easy to verify that the symmetric part of A is a symmetric matrix and that the
skew-symmetric part of A is a skew-symmetric matrix. The sum of the symmetric
part of A and the skew-symmetric part of A , A S A A ,is A : þ

1
2 ð

1
2 ð

A S A A A T A T A ¼ þ ¼ A þ Þþ A Þ: (A.15)

This result shows that any square matrix can be decomposed into the sum of a
symmetric and a skew-symmetric matrix or anti-symmetric matrix. Using the trace
operation introduced above, the representation (A.15) can be extended to three-way
decomposition of the matrix A ,

tr A
n

1
2

2 tr A
n

1
2 ð

A T A T A ¼ 1 þ A þ 1 þ A Þ: (A.16)