The DFT can be formulated as a complex matrix multiply, as we show in
this section. (This section can be omitted without affecting what
follows.) For basic definitions regarding matrices, see
Appendix H.

The DFT consists of inner products of the input signal
with sampled complex sinusoidal sections :

By collecting the DFT output samples into a column vector, we have

or

where
denotes the
DFT matrix
, i.e.,

The notation
denotes the
Hermitian transpose of the complex matrix (transposition
and complex conjugation).

Note that the th column of
is the th DFT sinusoid, so
that the th row of the DFT matrix
is the
complex-conjugate of the th DFT sinusoid. Therefore, multiplying
the DFT matrix times a signal vector
produces a column-vector
in which the th element
is the inner
product of the th DFT sinusoid with
, or
, as expected.

Computation of the DFT matrix in Matlab is illustrated in §I.4.3.

The inverse DFT matrix is simply
. That is,
we can perform the inverse DFT operation as

(6.1)

Since the forward DFT is
,
substituting
from Eq. (6.1) into the forward DFT
leads quickly to the conclusion that

(6.2)

This equation succinctly states that the columns of
are orthogonal, which, of course, we already knew. I.e.,
for , and
: