Positive definite iff eigenvalues are positive

Dependencies:

1. Positive definite
2. Eigenvalues and Eigenvectors
3. All eigenvalues of a hermitian matrix are real
4. Bounding matrix quadratic form using eigenvalues

Let $A$ be a real $n$ by $n$ symmetric matrix. This means that all its eigenvalues are real. Let $\lambda_{\textrm{min}}$ and $\lambda_{\textrm{max}}$ be the minimum and maximum eigenvalues. Then

• $A$ is positive definite iff $\lambda_{\textrm{min}} > 0$.
• $A$ is positive semidefinite iff $\lambda_{\textrm{min}} \ge 0$.
• $A$ is negative definite iff $\lambda_{\textrm{max}} < 0$.
• $A$ is negative semidefinite iff $\lambda_{\textrm{max}} \le 0$.

Proof

\[ \forall u \in \mathbb{R}^n, \lambda_{\textrm{min}}u^Tu \le u^TAu \le \lambda_{\textrm{max}}u^Tu \] Also, the above bounds are tight. The theorems of this page directly follow from the above bound.

Dependency for:

1. Bound on eigenvalues of sum of matrices

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• Number of transitive dependencies: 96

Transitive dependencies:

1. /linear-algebra/eigenvectors/cayley-hamilton-theorem
2. /misc/fundamental-theorem-of-algebra
3. /complex-numbers/complex-numbers
4. /linear-algebra/matrices/gauss-jordan-algo
5. /linear-algebra/vector-spaces/condition-for-subspace
6. /complex-numbers/conjugate-product-abs
7. /complex-numbers/conjugation-is-homomorphic
8. /sets-and-relations/equivalence-relation
9. /sets-and-relations/composition-of-bijections-is-a-bijection
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95. Orthogonally diagonalizable iff hermitian
96. Bounding matrix quadratic form using eigenvalues