By introducing the product topology on #math23#Rm×m×Rn×n with the induced inner product
#math24#
〈(A1, B1),(A2, B2)〉 : = 〈A1, A2〉 + 〈B1, B2〉,
(3)
we calculate the Fréchet derivative of F as follows:
#math25#
F'(U, V)(H, K)
=
〈R(U, V), HΣVT + UΣKT - P(HΣVT + UΣKT)〉
=
〈R(U, V), HΣVT + UΣKT〉
(4)
=
〈R(U, V)VΣT, H〉 + 〈ΣTUTR(U, V), KT〉.
In the middle line of (#eq2.11#111>) we have used the fact that the range of
R is always perpendicular to the range of P. The gradient ∇F of
F, therefore, may be interpreted as the
pair of matrices:
where #math28##tex2html_wrap_inline1278#(U, V)(#tex2html_wrap_inline1279#(m)×#tex2html_wrap_inline1280#(n)) stands for the
tangent space to the manifold #math29##tex2html_wrap_inline1282#(m)×#tex2html_wrap_inline1283#(n) at #math30#(U, V)∈#tex2html_wrap_inline1285#(m)×#tex2html_wrap_inline1286#(n) and so on. The projection of
#math31#∇F(U, V) onto #math32##tex2html_wrap_inline1291#(U, V)(#tex2html_wrap_inline1292#(m)×#tex2html_wrap_inline1293#(n)),
therefore, is the product of the projection of the first component of
#math33#∇F(U, V) onto #math34##tex2html_wrap_inline1298#U#tex2html_wrap_inline1299#(m) and the projection of the
second component of #math35#∇F(U, V) onto #math36##tex2html_wrap_inline1304#V#tex2html_wrap_inline1305#(n).
In particular, we claim that the
projection g(U, V) of the gradient #math37#∇F(U, V) onto
#math38##tex2html_wrap_inline1311#(U, V)(#tex2html_wrap_inline1312#(m)×#tex2html_wrap_inline1313#(n)) is given by the pair of
matrices:
defines a steepest descent flow on the manifold #math41##tex2html_wrap_inline1326#(m)×#tex2html_wrap_inline1327#(n) for the objective function F(U, V).