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Catalecticant Matrix

A catalecticant matrix is the matrix of a catalecticant map in appropriately normalized monomial bases. For the normalized binary form

F(x,y)=sum_(k=0)^d(d; k)a_kx^(d-k)y^k,

the rth catalecticant matrix, with a compatible normalization of the bases, has the Hankel matrix form

Cat_(r)(F)=(a_(i+j))_(0<=i<=d-r, 0<=j<=r).

It has d-r+1 rows and r+1 columns. Vanishing minors and bounds on its matrix rank give conditions for F to be expressible as a sum of powers of linear forms.

See also

Binary Form, Catalecticant, Hankel Matrix, Matrix Rank

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References

Iarrobino, A. and Kanev, V. Power Sums, Gorenstein Algebras, and Determinantal Loci. Berlin, Germany: Springer-Verlag, 1999.

Cite this as:

Weisstein, Eric W. "Catalecticant Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CatalecticantMatrix.html

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