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Why $[(\mathbf{I}_N-\mathbf{A}^\top \mathbf{A})\mathbf{x}]$ is Gaussian with i.i.d. Gaussian $\mathbf{A}$?
Partial answer. I hope I do not make mistakes. The vectors $A^\top Ax$ and $(I-A^\top Ax)$ are NOT gaussian, and my impression is that the gaussian character is just an asymptotic law as \$1 << M ...