Real Symmetric Positive Definite Matrix has Positive Eigenvalues

Theorem
Let $A$ be a symmetric positive definite matrix over $\mathbb R$.

Let $\lambda$ be an eigenvalue of $A$.

Then $\lambda$ is real with $\lambda > 0$.

Proof
Let $\lambda$ be an eigenvalue of $A$ and let $\mathbf v$ be a corresponding eigenvector.

From Real Symmetric Matrix has Real Eigenvalues, $\lambda$ is real.

From the definition of a positive definite matrix, we have:


 * $\mathbf v^\intercal A \mathbf v > 0$

That is:

From Euclidean Space is Normed Space, we have:


 * $\norm {\mathbf v}^2 > 0$

so:


 * $\lambda > 0$