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For real symmetric matrices, positive leading principal minors are equivalent to positive Cholesky pivots

Statement

Let $n \geq 1$ , let $A \in M_{n} (R)$ be symmetric, and let $Δ_{k}$ be the determinant of its leading $k \times k$ principal submatrix.

Then the following are equivalent:

$Δ_{k} > 0$ for every $1 \leq k \leq n$ .
$A$ has a Cholesky factorisation $A = L L^{T}$ with positive diagonal.

When $A = L L^{T}$ , one has

$Δ_{k} = \prod_{i < k} ℓ_{i i}^{2},$

so positivity of the leading principal minors is equivalent to positivity of the Cholesky pivots.

Facts & Assumptions

Given: A real symmetric matrix $A \in M_{n} (R)$ .

[L1]

A matrix has a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite (A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique).

[L2]

A real symmetric matrix is positive definite exactly when all of its leading principal minors are positive (Sylvester's criterion: a real symmetric $n \times n$ matrix with $n \geq 1$ is positive definite if and only if all leading principal minors are positive).

[L3]

The determinant of a triangular matrix is the product of its diagonal entries (The determinant of a triangular matrix is the product of its diagonal entries).

Proof

Proof technique: compare Sylvester's criterion with the determinant formula for leading Cholesky blocks.

1.1

By [L1] and [L2], condition 1 is equivalent to positive definiteness, and positive definiteness is equivalent to existence of a Cholesky factorisation with positive diagonal. Hence conditions 1 and 2 are equivalent.

L1L2

2.1

Suppose $A = L L^{T}$ with $L$ lower triangular and positive diagonal. Its leading $k \times k$ principal block factors as $A_{k} = L_{k} L_{k}^{T}$ , where $L_{k}$ is the leading $k \times k$ block of $L$ . By [L3], $Δ_{k} = \det (A_{k}) = \det (L_{k})^{2} = {(\prod_{i < k} ℓ_{i i})}^{2} = \prod_{i < k} ℓ_{i i}^{2} .$ Since each $ℓ_{i i} > 0$ , every $Δ_{k}$ is positive.

step 1.1L3algebra

3.1

Step 1.1 proves the equivalence, and step 2.1 identifies the leading principal minors with the squared pivot products.

step 1.1step 2.1∎

Depends on

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Section 2.9 (standard reference, not scraped)
David Bindel, CS 4220: Numerical Analysis, Beyond linear solves + using structure (standard reference, not scraped)