Characterisation of Real Symmetric Positive Definite Matrix/Necessary Condition

Theorem
Let $A$ be an $n \times n$ positive definite symmetric matrix over $\mathbb R$.

Then:


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

Proof
Let $A$ be positive definite.

From Real Symmetric Matrix is Orthogonally Diagonalizable:


 * there exists an orthogonal matrix $P$ and diagonal matrix $D$ such that $A = P^\intercal D P$.

Further:


 * the diagonal entries of $D$ are the eigenvalues of $A$.

From Real Symmetric Positive Definite Matrix has Positive Eigenvalues:


 * the diagonal entries of $D$ are positive.

We can therefore construct a real diagonal matrix $S$ by:


 * $\paren S_{i j} = \begin{cases}\sqrt {\paren D_{i i} } & i = j \\ 0 & i \ne j\end{cases}$

From Product of Diagonal Matrices is Diagonal, we have:


 * $\paren {S^2}_{i j} = \begin{cases}\paren D_{i i} & i = j \\ 0 & i \ne j\end{cases}$

so:


 * $S^2 = D$

We also have:

We therefore have:

So from Matrix is Invertible iff Determinant has Multiplicative Inverse:


 * $P^\intercal S P$ is invertible.

Let $C = P^\intercal S P$.

Then:

Since $C$ is invertible, we are done.