Sum of positive definite matrices is positive definite

Dependencies:

1. Positive definite

Let $A$ and $B$ be real $n$ by $n$ matrices. Then

• If $A$ and $B$ are positive definite, then $A+B$ is positive definite.
• If $A$ and $B$ are positive semidefinite, then $A+B$ is positive semidefinite.
• If $A$ and $B$ are negative definite, then $A+B$ is negative definite.
• If $A$ and $B$ are negative semidefinite, then $A+B$ is negative semidefinite.

Proof

If $A$ and $B$ are positive definite, \[ \forall u \in \mathbb{R}^n - \{0\}, u^T(A+B)u = u^TAu + u^TBu > 0 + 0 = 0 \] If $A$ and $B$ are positive semidefinite, \[ \forall u \in \mathbb{R}^n, u^T(A+B)u = u^TAu + u^TBu \ge 0 + 0 = 0 \] If $A$ and $B$ are negative definite, \[ \forall u \in \mathbb{R}^n - \{0\}, u^T(A+B)u = u^TAu + u^TBu < 0 + 0 = 0 \] If $A$ and $B$ are negative semidefinite, \[ \forall u \in \mathbb{R}^n, u^T(A+B)u = u^TAu + u^TBu \le 0 + 0 = 0 \]

Dependency for:

1. Bound on eigenvalues of sum of matrices

Info:

• Depth: 4
• Number of transitive dependencies: 5

Transitive dependencies:

1. Group
2. Ring
3. Semiring
4. Matrix
5. Positive definite