Physical Chemistry Third Edition

1250 B Some Useful Mathematics

If two matrices are square, they can be multiplied together in either order. However,

the multiplication is not always commutative. It is possible that

ABBA (in some cases) (B-86)

Matrix multiplication isassociative:

A(BC)(AB)C (B-87)

Matrix multiplication and addition aredistributive:

A(B+C)AB+AC (B-88)

Matrix multiplication is similar to operator multiplication in that both are associative

and distributive but not necessarily commutative.

We define an identity matrixEsuch that

EAAEA (B-89)

The symbolEis taken from the German wordEinheit(unity). The fact that we require

Eto be the identity matrix when multiplied on either side ofArequires bothAandE

to be square matrices, butEcan have any number of rows and columns. It has the form

E

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

100 ··· 0

010 ··· 0

001 ··· 0

··· ··· ··· ··· ···

000 ··· 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(B-90)

Thediagonal elementsof any square matrix are those with both indices equal. The

diagonal elements ofEare all equal to 1 and are the only nonzero elements. This can

be written in the form:

Eijδij

{

1ifij

0ifij

(B-91)

The quantityδijis called theKronecker deltaand is defined by the second equality.

Division by a matrix is not defined, but we define theinverseof a square matrix. We

denote the inverse ofAbyA−^1 and require that

A−^1 AAA−^1 E (B-92)

This multiplication of a matrix and its inverse is commutative, so thatAis the inverse

ofA−^1.

Associated with each square matrix is a determinant (see below). If the determinant

of a square matrix vanishes, the matrix is said to besingular. A singular matrix has no

inverse. Thetraceof a square matrix is the sum of the diagonal elements of the matrix:

Tr(A)

∑n

i 1 Aii (B-93)