# Sums of positive definite matrices are still positive definite

For a matrix $$A$$, the notation $$A\succeq0$$ and $$A\succ0$$ when $$A$$ is often used to denote positive semi-definite (PSD) or positive definite (PD) matrices, respectively. Using the definition of a PD matrix, we can prove that the sum of two PD matrices is also PD. A very similar approach can be used to prove the sum of two PSD matrices is also PSD.

$$\textbf{Proof:}$$ Let both $$A$$ and $$B$$ be $$d \times d$$ matrices. Also, let $$A \succ 0$$ and $$B \succ 0$$. Then by the definition of positive definite matrices, $$\textbf{x}^T A \textbf{x} > 0$$ and $$\textbf{x}^T B \textbf{x} > 0$$ for all non-zero vectors $$\textbf{x}$$.

Now, notice that $\textbf{x}^T (A + B) \textbf{x} = \textbf{x}^T A \textbf{x} + \textbf{x}^T B \textbf{x}$ by the matrix distributive property. Since we know that both $$\textbf{x}^T A \textbf{x} > 0$$ and $$\textbf{x}^T B \textbf{x} > 0$$, and that the sum of two positive numbers is also positive. Then, $\textbf{x}^T A \textbf{x} + \textbf{x}^T B \textbf{x} > 0.$ This implies that $\textbf{x}^T (A + B) \textbf{x} > 0.$ Thus, the sum of two positive definite matrices $$A$$ and $$B$$ is positive definite. $$\blacksquare$$ ##### Eric McKinney
###### PhD Student in Statistics
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