Question

Question 7: Prove that an \(n \times n\) A is positive definite if and only if A admits a Cholesky factorization, namely, \(A = {R^T}R\) for some invertible upper triangular matrix R whose diagonal entries are all positive. (Hint; Use a QR factorization and Exercise 26 in Section 7.2.)

Short answer

It is proved that a \(n \times n\) matrix A is positive definite if and only if A admits a Cholesky factorization.

Step-by-step solution

QR factorization

Theorem 12 in section 6.4states that when \(A\) is a \(m \times n\) matrix that islinearly independent columns, then \(A\) may be factored as \(A = QR\), with \(Q\) is an \(m \times n\) matrix wherein columns provide an orthonormal basisfor \({\mathop{\rm Col}\nolimits} A\), and \(R\) is an \(n \times n\) upper triangular invertible matrix which has positive entries on its diagonal.

Show that \(n \times n\) A is positive definite if and only if A admits a Cholesky factorization

Exercise 25 in section 7.2states that when \(B\) is a \(m \times n\) matrix then \({B^T}B\) ispositive semidefiniteand when \(B\) is a \(n \times n\) matrix then \({B^T}B\) ispositive definite.

Exercise 26 in section 7.2states that when an \(n \times n\) A matrix is positive definite, then there is a positive definite matrix \(B\) such that \(A = {B^T}B\).

When \(A\) permits a Cholesky factorization, namely \(A = {R^T}R\), with \(R\) is the upper triangular with positive diagonal entries then \(\det R = {r_{11}}{r_{22}} \cdots {r_{nn}} > 0\) and R is invertible.

Thus, A is a positive definite according to Exercise 25 in Section 7.2.

Assume that \(A\) is positive definite. Then \(A = {B^T}B\) for any positive definite matrix B, according to Exercise 26 in section 7.2. \(B\) is invertible and 0 is not an eigenvalue of \(B\) because its eigenvalues are positive.

Therefore, the columns of \(B\) are linearly independent. According to theorem 12 in section 6.4, \(B = QR\) for some \(n \times n\) matrix \(Q\) has orthonormal columns, and also some upper triangular matrix \(R\) has positive diagonal entries.

So, \({Q^T}Q = I\) because \(Q\) is a square matrix,

\(\begin{array}{c}A = {B^T}B\\ = {\left( {QR} \right)^T}\left( {QR} \right)\\ = {R^T}{Q^T}QR\\ = {R^T}R\end{array}\)

Therefore, \(R\) possess the necessary properties.

Hence, it is proved that a \(n \times n\) matrix A is positive definite if and only if A admits a Cholesky factorization.