1 | From (7) ∼ (9) set H_{b} and H_{w} as follows. H b = N s [ m ¯ 1 ( 1 ) − m ¯ t ( l ) , … , m ¯ D ( l ) − m ¯ t ( l ) ] , H w = [ A 1 ( 1 ) − m ¯ 1 ( 1 ) − 1 ¯ N s T , … , A D ( 1 ) − m ¯ D ( 1 ) 1 ¯ N s T ] . Here, 1 ¯ N s is the 1-vector of size N_{s}. Note that S_{b}^{(l)} =H_{b}H_{b}^{H} and S_{w}^{(l)}=H_{w}H_{w}^{H} are satisfied. |

2 | Compute the SVD(singular value decomposition) of matrix Z which is composed of (H_{b},H_{w}). Z = [ H b H H b H ] ∈ C D ( N s + 1 ) × M ( N − M ) , Z = P [ Λ 0 0 0 ] U H . Here, Λ is a diagonal matrix with effective rank s, P ∈ C D ( N s + 1 ) × D ( N s + 1 ) and U ∈ C M ( N − M ) × M ( N − M ) are orthogonal matrices. |

3 | Partition the matrix P as P = [ P 11 P 12 P 21 P 22 ] , where P_{11}C^{(D×s)}, P 12 ∈ C D × ( D ( N s + 1 ) − s ) , P 21 ∈ C D N s × s , and P 22 ∈ C D N s × ( D ( N s + 1 ) − s ) are submatrices. Compute the orthogonal matrix V from the SVD of P_{11}. P 11 = W ∑ V H . |

4 | Compute X as X = U [ Λ − 1 V 0 0 V ] . Then set transformation matrix G ¯ ( l ) as G ¯ ( l ) = [ [ X ] 1 , [ X ] 2 , … , [ X ] M D ] ∈ C M ( N − M ) × M D , where [X]_{i} is ith column of matrix X. |