80. Leading Principal Minor

A leading principal minor of order is the determinant of the top-left submatrix of .

That is, take rows and columns indexed — no other choices.

For a matrix, the three leading principal minors are:

Order 1: . Order 2: . Order 3: itself.

80.1. Sylvester’s criterion (positive-definiteness)

A symmetric matrix is positive definite iff all leading principal minors are strictly positive:

where is the order- leading principal minor.

For negative definite: signs alternate — (i.e. ).

Example

, . So is positive definite.

80.2. See also