DOA estimation; Music (Multiple Signal Classification) algorithm
The receive-signals covariance matrix $\mathbf R$ is by its definition 
$$\DeclareMathOperator{\expect}{E}\newcommand{\E}[1]{\expect\left[#1\right]}\DeclareMathOperator{\diag}{diag}$$
$$\mathbf R= \E{\mathbf{YY}^*}=\mathbf R^*$$ hermitian.

We don't just find any decomposition of $\mathbf R$ into subspaces – we pick the Eigenvalue Decomposition, which gives us

$$\mathbf R= \mathbf U \mathbf \Lambda \mathbf U^* \text,$$

where 

\begin{align}
\mathbf\Lambda &amp;= \diag(\lambda_1,\ldots,\lambda_N) &amp;\text{diagonal matrix of Eigenvalues,}\\
\mathbf U &amp;= ( \mathbf g_1,\ldots, \mathbf g_N)&amp;\text{matrix: Eigenvectors as columns}
\end{align}

We can without loss of generality have $\mathbf \Lambda$ sorted such that $\lambda_1,\ldots\lambda_M$ and thus $\mathbf g_1,\ldots \mathbf g_M$ correspond to transmit signals, and $\lambda_{M+1},\ldots,\lambda_N$ and $\mathbf g_{M+1},\ldots,\mathbf g_N$ come from to noise.

Assuming our receive array is well-formed and the signal is not incident from an angle that leads to ambiguities, $\lambda_i\ne\lambda_{l\ne i}$.
Now, in the presence of uncorrelated noise, we know that


$\mathbf R$ has full rank, and 
since $\mathbf R$ is hermitian, different Eigenvalues have orthogonal Eigenvectors. 


So, if we have $M&lt;N$ independent receive signal components, we get $M$ different, orthogonal vectors.

The $M+1,\ldots, N$th eigenvalues of $\mathbf R$ are the noise variance $\sigma^2\ne\lambda_i,\,\forall i\in\left\{1,\ldots,M\right\}$, we see that the space spanned by the signal-caused Eigenvector must be orthogonal to the noise subspace.