Chió Pivotal Condensation -- from Wolfram MathWorld

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Chió Pivotal Condensation

Chió pivotal condensation is a method for evaluating an determinant in terms of 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 matrix with by forming the matrix such that

(1)

Then

(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).

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.

CITE THIS AS:

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