How to prove Wielandt minimax formula?

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The statements are as follows:
Let $1leqslant i_1<i_2<cdots<i_kleqslant n$ be integers. Define a partial flag to be a nested collection $V_1subset V_2cdotssubset V_k$ of subspaces of $mathbbC^n$ s.t. $dim(V_j)=i_j$ for all j between 1 and k. Define the Schubert variety $X(V_1,...,V_k)$ to be the collection of all k-dimensional subspaces W s.t. $dim(Wcap V_j)geqslant j$. Show that for any Hermitian matrix A,
$$lambda_i_1+...+lambda_i_k=sup_V_1,...,V_k inf_Win X(V_1,...,V_k)tr(A|_W)$$.
Where $lambda_i_j$ means the $i_j$ th eigenvalue of A, from large to small. The trace in the right formula stands for the partial trace of A on W.



I have proofed that the LHS is no bigger than the RHS using induction on k, but I do not know how to prove the other side.




linear-algebra eigenvalues-eigenvectors random-matrices




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    The statements are as follows:
    Let $1leqslant i_1<i_2<cdots<i_kleqslant n$ be integers. Define a partial flag to be a nested collection $V_1subset V_2cdotssubset V_k$ of subspaces of $mathbbC^n$ s.t. $dim(V_j)=i_j$ for all j between 1 and k. Define the Schubert variety $X(V_1,...,V_k)$ to be the collection of all k-dimensional subspaces W s.t. $dim(Wcap V_j)geqslant j$. Show that for any Hermitian matrix A,
    $$lambda_i_1+...+lambda_i_k=sup_V_1,...,V_k inf_Win X(V_1,...,V_k)tr(A|_W)$$.
    Where $lambda_i_j$ means the $i_j$ th eigenvalue of A, from large to small. The trace in the right formula stands for the partial trace of A on W.



    I have proofed that the LHS is no bigger than the RHS using induction on k, but I do not know how to prove the other side.




    linear-algebra eigenvalues-eigenvectors random-matrices




    share|cite|improve this question























      up vote
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      down vote

      favorite









      up vote
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      The statements are as follows:
      Let $1leqslant i_1<i_2<cdots<i_kleqslant n$ be integers. Define a partial flag to be a nested collection $V_1subset V_2cdotssubset V_k$ of subspaces of $mathbbC^n$ s.t. $dim(V_j)=i_j$ for all j between 1 and k. Define the Schubert variety $X(V_1,...,V_k)$ to be the collection of all k-dimensional subspaces W s.t. $dim(Wcap V_j)geqslant j$. Show that for any Hermitian matrix A,
      $$lambda_i_1+...+lambda_i_k=sup_V_1,...,V_k inf_Win X(V_1,...,V_k)tr(A|_W)$$.
      Where $lambda_i_j$ means the $i_j$ th eigenvalue of A, from large to small. The trace in the right formula stands for the partial trace of A on W.



      I have proofed that the LHS is no bigger than the RHS using induction on k, but I do not know how to prove the other side.




      linear-algebra eigenvalues-eigenvectors random-matrices




      share|cite|improve this question













      The statements are as follows:
      Let $1leqslant i_1<i_2<cdots<i_kleqslant n$ be integers. Define a partial flag to be a nested collection $V_1subset V_2cdotssubset V_k$ of subspaces of $mathbbC^n$ s.t. $dim(V_j)=i_j$ for all j between 1 and k. Define the Schubert variety $X(V_1,...,V_k)$ to be the collection of all k-dimensional subspaces W s.t. $dim(Wcap V_j)geqslant j$. Show that for any Hermitian matrix A,
      $$lambda_i_1+...+lambda_i_k=sup_V_1,...,V_k inf_Win X(V_1,...,V_k)tr(A|_W)$$.
      Where $lambda_i_j$ means the $i_j$ th eigenvalue of A, from large to small. The trace in the right formula stands for the partial trace of A on W.



      I have proofed that the LHS is no bigger than the RHS using induction on k, but I do not know how to prove the other side.





      linear-algebra eigenvalues-eigenvectors random-matrices





      share|cite|improve this question












      share|cite|improve this question




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      edited Jul 27 at 0:35
























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