Characterisation of Real Symmetric Positive Definite Matrix/Sufficient Condition

Theorem

Let $A$ be an $n \times n$ symmetric matrix over $\mathbb R$ such that:

there exists a nonsingular matrix $C$ such that $A = C^\intercal C$.

Then $A$ is positive definite.

Proof

Let $A$ be a symmetric matrix such that:

there exists an nonsingular matrix $C$ such that $A = C^\intercal C$.

Let $\mathbf v$ be a non-zero vector.

Then:

\(\ds \mathbf v^\intercal A \mathbf v\)

\(=\)

\(\ds \mathbf v^\intercal C^\intercal C \mathbf v\)

\(\ds \)

\(=\)

\(\ds \paren {C \mathbf v}^\intercal C \mathbf v\)

Transpose of Matrix Product

\(\ds \)

\(=\)

\(\ds \paren {C \mathbf v} \cdot \paren {C \mathbf v}\)

Definition of Dot Product

\(\ds \)

\(=\)

\(\ds \norm {C \mathbf v}^2\)

Dot Product of Vector with Itself

\(\ds \)

\(>\)

\(\ds 0\)

Euclidean Space is Normed Vector Space

So $A$ is positive definite.

$\blacksquare$