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Chió Pivotal Condensation

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Chió pivotal condensation is a method for evaluating an n×n determinant in terms of (n-1)×(n-1) determinants. It also leads to some remarkable determinant identities (Eves 1996, p. 130). Chiío's pivotal condensation is a special case of Sylvester's determinant identity.

Chió's condensation is carried out on an n×n matrix A=[a_(ij)] with a_(ii)!=0 by forming the (n-1)×(n-1) matrix B=[b_(ij)] such that

b_(ij)=a_(1,1)a_(i+1,j+1)-a_(1,j+1)a_(i+1,1).
(1)

Then

det(A)=(det(B))/(a_(11)^(n-2)).
(2)

Explicitly,

det(A)=1/(a_(11)^(n-2))||a_(11) a_(12); a_(21) a_(22)| |a_(11) a_(13); a_(21) a_(23)| ... |a_(11) a_(1n); a_(21) a_(2n)|; |a_(11) a_(12); a_(31) a_(32)| |a_(11) a_(13); a_(31) a_(33)| ... |a_(11) a_(1n); a_(31) a_(3n)|; | | ... |; |a_(11) a_(12); a_(n1) a_(n2)| |a_(11) a_(13); a_(n1) a_(n3)| ... |a_(11) a_(1n); a_(n1) a_(nn)||
(3)

(Eves 1996, pp. 129-134).

See also

Condensation, Determinant, Determinant Expansion by Minors, Sylvester's Determinant Identity

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References

Chió, F. "Mémoire sur les fonctions connues sous le nom de résultantes ou de déterminants." Turin: E. Pons, 1853.Eves, H. "Chio's Expansion." §3.6 in Elementary Matrix Theory. New York: Dover, pp. 129-136, 1996.Householder, A. S. The Theory of Matrices in Numerical Analysis. New York: Dover, 1975.Kahan, W. "Chió's Trick for Linear Equations with Integer Coefficients." http://www.cs.berkeley.edu/~wkahan/MathH110/chio.pdf.

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Chió Pivotal Condensation

Cite this as:

Weisstein, Eric W. "Chió Pivotal Condensation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChioPivotalCondensation.html

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