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Symmetric functions

Prove and interpret the matrix form of Lagrange's identity

Lagrange identity {$det(F^T F) = ∑_{|P |=m} det^2(F_P)$} which holds for all n × m matrices, where the sum to the right is over all m × m sub-matrices P of F, a formula which in calculus becomes for vectors {$|\vec{v}|^2|\vec{w}|^2 − (\vec{v}·\vec{w})2 = |\vec{v} ∧ \vec{w}|^2$}.

• How does the theory of symmetric functions of the eigenvalues of a matrix work in the case of generalized eigenvalues and Jordan canonical form?
• Consider the symmetric functions of the eigenvalues of a matrix in the case of a polar decomposition of a matrix.
• The combinatorics of symmetric functions of the eigenvalues of a matrix is all in terms of circular loops. How is that related to the fundamental theorem of covering spaces, the enumeration of equivalences as loops, as in homotopy theory?