Properties of Unitary transforms :(contd.)

3) Decorrelation :

When input vector elements are highly correlated , the transform coefficients tend to be uncorrelated. That is , the off-diagonal terms of covariance matrix tend to be small compared to diagonal elements.

4) Other properties : (i) The determinant and eigenvalues of a unitary matrix have unity magnitudte.

(ii) Entropy of a random vector is observed under unitary transformation.Since entropy is a measure of average information of the random vector, this means information is preserved under unitary transformation.

Example :

Given the entropy of an Gaussian random vector with mean and covariance , as :

=

To show is invariant under any unitary transformation.

Let =

= =

=

Use the definition of , we have

= =

Now

	=

	=

	=

Hence

= is invariant under any unitary transformation.