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3.19 Discrete Cosine Transforms 103

The transform matrix can, after simplification, be written

The odd rows (basis vectors) are symmetric and the even rows are antisym-
metric. Now, the inner product

of two vectors is zero if the vectors are orthogonal. We can simultaneously compute
several inner products using matrices. Hence, T T ^ is a matrix with only non-zero
element values on the diagonal if the row vectors are orthogonal and a unit matrix
if the row vectors are orthonormal. Obviously, the basis vectors are, in our case,
non-orthogonal. However, the even and odd rows are mutually orthogonal.

EXAMPLE 3.9

Show that the two-dimensional discrete cosine transform, which is defined
shortly, can be computed by using only one-dimensional DCTs. What other types