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Minor


Minor

A minor M_(ij) is the reduced determinant of a determinant expansion that is formed by omitting the ith row and jth column of a matrix A. So, for example, the minor M_(22) of the above matrix is given by

 M_(22)=|a_(11) a_(13) a_(14) ... a_(1n); a_(31) a_(33) a_(34) ... a_(3n); a_(41) a_(43) a_(44) ... a_(4n); | | | ... |; a_(k1) a_(k3) a_(k4) ... a_(kn)|.

The (i,j)th minor can be computed in the Wolfram Language using

  Minor[m_List?MatrixQ, {i_Integer, j_Integer}] :=
    Det[Drop[Transpose[Drop[Transpose[m], {j}]],
      {i}]]

The Wolfram Language's built-in Minors[m] command instead gives the minors of a matrix m obtained by deleting the (n-i+1)st row and (n-j+1)st column of m, while Minors[m, k] gives the kth minors of m. The Minor code above therefore corresponds to (i,j)th entry of

  MinorMatrix[m_List?MatrixQ] := Map[Reverse,
    Minors[m], {0, 1}]

i.e., the definition Minors[m, {i, j}] is equivalent to MinorMatrix[m][[i, j]].


See also

Cofactor, Determinant, Determinant Expansion by Minors

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 169-170, 1985. Lichtblau, D. "Symbolic FAQ." http://library.wolfram.com/infocenter/Conferences/325.Muir, T. "Minors and Expansion." Ch. 4 in A Treatise on the Theory of Determinants. New York: Dover, pp. 53-137, 1960.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 235, 1990.

Referenced on Wolfram|Alpha

Minor

Cite this as:

Weisstein, Eric W. "Minor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Minor.html

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