Positive Semidefinite Matrix

Last updated Nov 1, 2022

Suppose $A \in \mathbb{R}^{n \times n}$ for some $n \in \mathbb{N}$ and that $\forall x \in \mathbb{R}^{n}$, $$x^{T} A x \geq 0$$. Then $A$ is a Positive Semidefinite Matrix. We denote the set of $n \times n$ Positive Semidefinite Matrixs $\mathbb{S}_{n}^{+}$

1. A Square Matrix is Positive Semidefinite iff its Eigenvalues are Nonnegative
2. A Square Matrix is Positive Semidefinite iff all of its Principal Minors are Nonnegative
3. A Square Matrix is Positive Semidefinite iff it is the Gram Matrix of n vectors
4. A Square Matrix is Positive Semidefinite iff it is the product of a Matrix with its Transpose
5. Positive Semidefinite Matrices form a Convex Cone
6. Positive Semidefnite Matrices are Symmetric - Thus justifies the notation.
7. A Symmetric Matrix is Positive Semidefinite iff its Frobenius Inner Product with any Positive Semidefinite Matrix is Nonnegative

• Transpose Matrix
• Real Numbers
• Natural Numbers

• CMU Class Lecture 12
• Boyd - Convex Optimization
• Two Basic Probabilistic Proofs