Exercise 4.1.

Prove that a symmetric m m matrix A is positive semidefinite if and only if all its principal minors (i.e., determinants of square submatrices symmetric with respect to the diagonal) are nonnegative.

Hint. Look at the angular minors of the matrices A + ? I _n for small positive ?.

Demonstrate by an example that nonnegativity of angular minors of a symmetric matrix is not sufficient for the positive semidefiniteness of the matrix.

Exercise 4.2.

Diagonal-dominant matrices. Let a symmetric matrix A = [a _ij] ^m _{i,j =1} satisfy the relation

Prove that A is positive semidefinite.

Diagonalization

Exercise 4.3.

Prove the following standard facts from linear algebra:

1. If A is a symmetric positive semidefinite m m matrix and P is an n m matrix, then the matrix P AP ^T is positive semidefinite.

2. A symmetric m m matrix A is positive semidefinite if and only if it can be represented as A = Q ?Q ^T, where Q is orthogonal (Q ^TQ = I) and ? is diagonal with nonnegative diagonal entries. What are these entries? What...