I Have read that for a matrix of reals Y and a p.s.d matrix B that theMaximum of f ( Y ) = T r ( Y T B Y ) subject to Y T Y = I is achieved when s p a n ( Y ) equals the span of the first d eigen-vectors of B.What is the reasoning behind this eigen-solution leading to the maximum?

Question

I Have read that for a matrix of reals   Y and a p.s.d matrix   B that theMaximum of   f  (  Y  )  =  T  r  (      Y    T    B  Y  ) subject to       Y    T    Y  =  I is achieved when   s  p  a  n  (  Y  ) equals the span of the first   d eigen-vectors of   B.What is the reasoning behind this eigen-solution leading to the maximum?

Kaydence Washington · Accepted Answer

t    r          Y    T    B  Y  =      t    r      B  Y      Y    T  , and   Y      Y    T   is a rank-  d orthogonal projection; any such projection is obtained in this way. So you maximize       t    r      B  P among rank-  d orthogonal projections   P, so it&#039;s a consequence of the Courant minimax principle for the eigenvalues, or of the Cauchy interlacing theorem

I Have read that for a matrix of reals Y and a p.s.d matrix B that the Maximum of

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