79. Principal Minor

A principal minor of an matrix is the determinant of a square submatrix whose diagonal is a subset of ‘s diagonal.

To form a principal minor: pick a subset of indices , then delete every row and column not in . The resulting submatrix is square; its determinant is the principal minor.

Pick rows/columns :

79.1. Levels (order)

A principal minor formed from indices is called a level- (or order-) principal minor.

For an matrix:

• There are levels (from to )
• The number of level- principal minors is
• Total number of principal minors: (including the trivial empty one)

79.2. Where they show up

• Sylvester’s criterion (positive-definiteness test): a symmetric matrix is positive definite iff every leading principal minor is positive
• Positive semi-definiteness: requires every principal minor (not just leading) to be non-negative
• Convex analysis — checking definiteness of the Hessian via principal minors

79.3. See also

• Leading Principal Minor — minors from the top-left corner
• Minor — general minors (rows and columns can differ)
• Determinant