Minors

Minors[m]

gives the minors of a matrix m.

Minors[m,k]

gives the k ^(th) minors.

Details

  • For an × matrix the ^(th) element of Minors[m] gives the determinant of the matrix obtained by deleting the ^(th) row and the ^(th) column of m.
  • Map[Reverse,Minors[m],{0,1}] makes the ^(th) element correspond to deleting the ^(th) row and ^(th) column of m.
  • Minors[m] is equivalent to Minors[m,n-1].
  • Minors[m,k] gives the determinants of the k×k submatrices obtained by picking each possible set of k rows and k columns from m.
  • Each element in the result corresponds to taking rows and columns with particular lists of positions. The ordering of the elements is such that reading across or down the final matrix, the successive lists of positions appear in lexicographic order.
  • For an × matrix Minors[m,k] gives an × matrix.
  • Minors[m,k,f] applies the function f rather than Det to each of the submatrices picked out.

Examples

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Basic Examples  (2)

Minors of a 3×3 matrix:

2×2 minors of a 2×3 matrix:

Scope  (2)

Minors of a 4×4 matrix:

The matrices used to compute the minors:

2×2 minors of a 3×4 matrix:

The matrices used to compute the minors:

Applications  (3)

Define the adjoint of a matrix:

Show its relation to Det and Inverse:

Compute the rank of a matrix:

The rank is the largest k for which Minors[mat,k] contains nonzero entries:

Use MatrixRank to check that the rank is indeed equal to two:

Find the singularities of an algebraic space curve:

The curve is the intersection of two surfaces:

Generate an explicit parametrization of the curve:

Properties & Relations  (5)

Minors of dimension higher than the MatrixRank are zero:

The Dimensions of the ^(th) minors of an × matrix are ×:

Minors of an IdentityMatrix gives an IdentityMatrix:

Minors of a DiagonalMatrix gives a DiagonalMatrix:

Minors is effectively equivalent to an outer product of row and column extractions:

Neat Examples  (1)

Some minor variations:

Introduced in 1988
 (1.0)
 |
Updated in 1999
 (4.0)
2000
 (4.1)
2002
 (4.2)