The transpose of a matrix

8.5 FUNCTIONS OF MATRICES
The identity matrix I has the property
AI = IA = A.
It is clear that, in order for the above products to be defined, the identity matrix
must be square. The N × N identity matrix (often denoted by IN ) has the form


1 0 ··· 0

.. 
 0 1
. 
.
IN = 
 .

.
.
.
 .
. 0 
0 ···
0
1
8.5 Functions of matrices
If a matrix A is square then, as mentioned above, one can define powers of A in
a straightforward way. For example A2 = AA, A3 = AAA, or in the general case
An = AA · · · A
(n times),
where n is a positive integer. Having defined powers of a square matrix A, we
may construct functions of A of the form
an An ,
S=
n
where the ak are 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 the exponential of a matrix, which is defined by
exp A =
∞
An
n=0
n!
.
(8.38)
This definition can, in turn, be used to define other functions such as sin A and
cos A.
8.6 The transpose of a matrix
We have seen that the components of a linear operator in a given coordinate system can be written in the form of a matrix A. We will also find it useful, however,
to consider the different (but clearly related) matrix formed by interchanging the
rows and columns of A. The matrix is called the transpose of A and is denoted
by AT .
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