# Mixed Alternating Projections with Application to Hankel Low-Rank Approximation
## 2 Alternating Projections
: _Using the commutativity property \(\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}}\mathbf{\Pi}_{\mathcal{A}}=\mathbf{ \Pi}_{\mathcal{A}}\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}}=\mathbf{\Pi}_{ \mathcal{A}\cap\mathcal{B}}\) and \(\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}}\mathbf{\Pi}_{\mathcal{B}}=\mathbf{ \Pi}_{\mathcal{B}}\mathbf{\Pi}_{\mathcal{A}\cap\mathcal{B}}=\mathbf{\Pi}_{ \mathcal{A}\cap\mathcal{B}}\), the expression \(\mathbf{P}_{\mathcal{A},\mathcal{B}}^{n}-\mathbf{\Pi}_{\mathcal{A}\cap \mathcal{B}}\) on the left side of the inequality (9) can be rewritten as_  
\[\mathbf{P}_{\mathcal{A},\mathcal{B}}^{n}-\mathbf{\Pi}_{\mathcal{A}\cap \mathcal{B}}=\mathbf{P}_{\mathcal{A},\mathcal{B}}^{n},\]  
_where \(\mathbf{P}_{\mathcal{A},\mathcal{B}}=\mathbf{\Pi}_{\mathcal{A}}\mathbf{\Pi}_{ \mathcal{B}}\), \(\mathbf{\Pi}_{\mathcal{A}}=\mathbf{\Pi}_{\mathcal{A}}-\mathbf{\Pi}_{\mathcal{A} \cap\mathcal{B}}\), \(\mathbf{\Pi}_{\mathcal{B}}=\mathbf{\Pi}_{\mathcal{B}}-\mathbf{\Pi}_{\mathcal{A} \cap\mathcal{B}}\), \(\mathcal{\widehat{A}}\) and \(\mathcal{\widehat{B}}\) are orthogonal complements of the \(\mathcal{A}\cap\mathcal{B}\) in the subspaces \(\mathcal{A}\) and \(\mathcal{B}\), respectively. Then \(\hat{\lambda}\) is the largest absolute eigenvalue of the matrix \(\mathbf{P}_{\mathcal{A},\mathcal{B}}\)._  
Let us show the relation of \(\mathbf{P}_{\mathcal{A},\mathcal{B}}\) to canonical correlations and principal angles between subspaces. **Proposition 2**. : _Let \(\mathbf{A}\) be the matrix consisting of a basis of the subspace \(\mathcal{A}\) as its columns and \(\mathbf{B}\) be the matrix consisting of a basis of the subspace \(\mathcal{B}\) as its columns. Then, the following is true:_  
\[\mathbf{P}_{\mathcal{A},\mathcal{B}}=\Big{(}\mathbf{B}^{\mathrm{T}}\mathbf{B} \Big{)}^{-1}\Big{(}\mathbf{B}^{\mathrm{T}}\mathbf{A}\Big{)}\Big{(}\mathbf{A}^{ \mathrm{T}}\mathbf{A}\Big{)}^{-1}\Big{(}\mathbf{A}^{\mathrm{T}}\mathbf{B} \Big{)}. \tag{10}\]  
Proof. : The formula (10) is obtained by multiplying the solutions of two least-squares approximation problems related to the subspaces \(\mathcal{A}\) and \(\mathcal{B}\). **Remark 2**. : _The matrix on the right side of the equality (10) arises in the problem of computing canonical correlations. The squares of canonical correlations are equal to the squares of cosines of the so-called principal angles between subspaces \(\mathcal{A}\) and \(\mathcal{B}\), spanned by the columns of the matrices, and are calculated using the eigenvalues of this matrix._Since we need only the leading principal angle, which is the angle between subspaces, we will use the simpler definition of the cosine of the angle between subspaces through the inner product. **Definition 1** ([18]). : _Let \(\mathcal{A}\) and \(\mathcal{B}\) be two linear subspaces of a Hilbert space with inner product \(\langle\cdot,\cdot\rangle\) and the corresponding (semi-)norm \(\|\cdot\|\). Then, the cosine of the angle between \(\mathcal{A}\) and \(\mathcal{B}\) is defined as_  
\[\cos(\mathcal{A},\mathcal{B})=\max_{A\in\mathcal{A},\mathcal{B}\in\mathcal{B} }\biggl{(}\frac{|\langle A,B\rangle|}{\|A\|\|B\|}\biggr{)}.\]