Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 395

B 11 B 12
B 21 B 22

A 11 A 12
A 21 A 22

A 11 B 11 þ A 12 B 21 A 11 B 12 þ A 12 B 22

A B ¼ ¼ ;

A 21 B 11 þ A 22 B 21 A 21 B 12 þ A 22 B 22

(A.22)

where, in this case, the products (A.20) and (A.21) stand for the n 2 2 2 4

separate equations, the right-hand sides of which are the four elements of the last
matrix in (A.22).

¼ ¼

The Matrix Transpose Indicial Notation

The dot between the matrix product AB indicates that one index from A and one
index from B is to be summed over. The positioning of the summation index on the
two matrices involved in a matrix product is critical and is reflected in the matrix
notation by the transpose of the matrix. In the three equations below, (A.21), study
carefully how the positions of the summation indices within the summation sign
change in relation to the position of the transpose on the matrices in the associated
matrix product:

B T A T A T B T ð A Þ ij ¼ A ik B jk ; ð B Þ ij ¼ A ki B kj ; ð Þ ij ¼ A ki B jk : (A.23)

The Linear Transformation

A system of linear equations representing a linear transformation ,

r 1 ¼ A 11 t 1 þ A 12 t 2 þþ A 1n t n ;

r 2 ¼ A 21 t 1 þ A 22 t 2 þþ A 2n t n ;

r n ¼ A n1 t 1 þ A n2 t 2 þþ A nn t n (A.36)

may be contracted horizontally using the summation symbol, thus

r 1 ¼ A 1k t k ;

r 2 ¼ A 2k t k ;

...

r n ¼ A nk t k : (A.37)