Simple case

If the matrix $\boldsymbol{X}$ is an orthogonal (or unitary in case of complex values) matrix, we have

$$\boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{X}\boldsymbol{X}^T = \boldsymbol{I}.$$

In this case the matrix $\boldsymbol{A}$ becomes

$$\boldsymbol{A}=\boldsymbol{X}\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T)=\boldsymbol{I},$$

and we have the obvious case

$$\boldsymbol{\epsilon}=\boldsymbol{y}-\tilde{\boldsymbol{y}}=0.$$

This serves also as a useful test of our codes.

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