A square matrix
( for the ith column vector
of ) is *unitary* if its inverse is equal to its conjugate transpose,
i.e., . In particular, if a unitary matrix is real ,
then and it is *orthogonal*. Both the column and row vectors
(
) of a unitary or orthogonal matrix are orthogonal
(perpendicular to each other) and normalized (of unit length), or
*orthonormal*, i.e., their inner product satisfies:

These orthonormal vectors can be used as the basis vectors of the n-dimensional vector space.

Any unitary (orthogonal) matrix can define a *unitary (orthogonal)
transform* of a vector
:

or in tabular form:

The first equation above is the **forward transform** and can be written
in component form as:

This is the transform coefficient representing the projection of vector onto the ith column vector of the transform matrix (for the ith basis of the n-dimensional space).

The second equation is the **inverse transform** and can be written in
component form as:

This equation represents the signal vector as a linear combination (weighted sum) of the column vectors of the transform matrix . Geometrically, is a vector in the n-dimensional space spanned by the orthonormal vectors as the bases, and each coefficient (coordinate) is the projection of onto the corresponding basis vector .

As the n-dimensional space can be spanned by the column vectors of
*any* n by n unitary (orthogonal) matrix, a vector in the space
can be represented by any of such matrices, each defining a different
transform.

**Examples:**

- The identical matrix
is orthogonal and
defines a special orthogonal transform . Here the ith column or
row vector of is
with the ith
element equal 1 and all others 0. The inverse transform is

In this special case, the signal and its transform are identical. - When the (m,n)th component of matrix is
,
the corresponding transform is discrete Fourier transform. The nth column
vector of the transform matrix represents a sinusoid of
a frequency , and the corresponding complex coordinate
represents the magnitude and phase of this nth
frequency component. As any two orthogonal coordinate systems can be
related by a rotation, Fourier transform represents the signal
in a rotated coordinate system.

An orthogonal ( unitary) transform (where ) can be
interpreted geometrically as the rotation of the vector about the origin,
or equivalently, the representation of the same vector in a rotated coordinate
system. A orthogonal (unitary) transform does not change the vector's
length:

as . This is the Parseval's relation. If is interpreted as a signal, then its length represents the total energy or information contained in the signal, which is preserved during any unitary transform.