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Found in: Page 395

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# 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.)

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

See the step by step solution

## Step 1: QR factorization

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

## Step 2: Show that $$n \times n$$ A is positive definite if and only if A admits a Cholesky factorization

Exercise 25 in section 7.2 states that when $$B$$ is a $$m \times n$$ matrix then $${B^T}B$$ is positive semidefinite and when $$B$$ is a $$n \times n$$ matrix then $${B^T}B$$ is positive definite.

Exercise 26 in section 7.2 states 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.