Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.5 FUNCTIONS OF MATRICES

TheidentitymatrixIhas the property

AI=IA=A.

It is clear that, in order for the above products to be defined, the identity matrix

must be square. TheN×Nidentity matrix (often denoted byIN)hastheform

IN=













10 ··· 0

01

..

.

..

.

..

. 0
0 ··· 01













.

8.5 Functions of matrices

If a matrixAissquarethen, as mentioned above, one can definepowersofAin

a straightforward way. For exampleA^2 =AA,A^3 =AAA, or in the general case

An=AA···A (ntimes),

wherenis a positive integer. Having defined powers of a square matrixA,we

may constructfunctionsofAof the form

S=

∑

n

anAn,

where theakare simple scalars and the number of terms in the summation may

be finite or infinite. In the case where the sum has an infinite number of terms,

the sum has meaning only if it converges. A common example of such a function

is theexponentialof a matrix, which is defined by

expA=

∑∞

n=0

An

n!

. (8.38)

This definition can, in turn, be used to define other functions such as sinAand

cosA.

8.6 The transpose of a matrix

We have seen that the components of a linear operator in a given coordinate sys-

tem can be written in the form of a matrixA. We will also find it useful, however,

to consider the different (but clearly related) matrix formed by interchanging the

rows and columns ofA. The matrix is called thetransposeofAand is denoted

byAT.