Characterisation of Real Symmetric Positive Definite Matrix/Sufficient Condition

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Theorem

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

there exists an invertible 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 invertible 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.

$\blacksquare$