7.6 POSITIVE DEFINITE MATRICES

Since the symmetric structure of a matrix forces its eigenvalues to be real, what additional property will force all eigenvalues to be positive (or perhaps just nonnegative)? To answer this, let's deal with real-symmetric matrices-the hermitian case follows along the same lines. If A ? R ^{n n} is symmetric, then, as observed above, there is an orthogonal matrix P such that A = PDP ^T, where D = diag ( ? ₁, ? ₂,..., ? _n) is real. If ? _i ? 0 for each i, then D ^1/2 exists, so

and ? _i > 0 for each i if and only if B is nonsingular. Conversely, if A can be factored as A = B ^T B, then all eigenvalues of A are nonnegative because for any eigenpair ( ?, x),

Moreover, if B is nonsingular, then N ( B) = 0 ? Bx ? 0, so ? > 0. In other words, a real-symmetric matrix A has nonnegative eigenvalues if and only if A can be factored as A = B ^T B, and all eigenvalues are positive if and only if B is nonsingular. A symmetric matrix A whose eigenvalues are positive is called positive definite, and when the eigenvalues are just nonnegative, A is said to be positive...