Hessian-Calculus

Lecture 5 Principal Minors and the Hessian

Eivind Eriksen

BI Norwegian School of Management Department of Economics

October 01, 2010

Eivind Eriksen (BI Dept of Economics)

Lecture 5 Principal Minors and the Hessian

October 01, 2010

1 / 25

Principal minors

Principal minors

Let A be a symmetric n × n matrix. We know that we can determine the definiteness of A by computing its eigenvalues. Another method is to use the principal minors. Definition A minor of A of order k is principal if it is obtained by deleting n − k rows and the n − k columns with the same numbers. The leading principal minor of A of order k is the minor of order k obtained by deleting the last n − k rows and columns. For instance, in a principal minor where you have deleted row 1 and 3, you should also delete column 1 and 3. Notation

n We write Dk for the leading principal minor of order k. There are k principal minors of order k, and we write ∆k for any of the principal minors of order k.

Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, 2010 2 / 25

Principal minors

Two by two symmetric matrices

Example

a Let A = b b be a symmetric 2 × 2 matrix. Then the leading principal c minors are D1 = a and D2 = ac − b 2 . If we want to find all the principal minors, these are given by ∆1 = a and ∆1 = c (of order one) and ∆2 = ac − b 2 (of order two).

Let us compute what it means that the leading principal minors are positive for 2 × 2 matrices: Example

a Let A = b b be a symmetric 2 × 2 matrix. Show that if D1 = a > 0 and c D2 = ac − b 2 > 0, then A is positive definite.

Eivind Eriksen (BI Dept of Economics)

Lecture 5 Principal Minors and the Hessian

October 01, 2010

3 / 25

Principal minors

Leading principal minors: An example

Solution If D1 = a > 0 and D2 = ac − b 2 > 0, then c > 0 also, since ac > b 2 ≥ 0. The characteristic equation of A is λ2 − (a + c)λ + (ac − b 2 ) = 0 and it has two solutions (since A...