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Positive definite matrix

Let $A\in {ℝ}^{n\times n}$ be a symmetric real matrix. $A$ is positive definite if for any nonzero vector $x\in {ℝ}^{n\times 1}$, $x^{\top }Ax>0$. If the inequality holds weakly, we say that $A$ is positive semidefinite.

Properties of Positive Definite Matrices

The following statements are equivalent:

• $x^{\top }Ax>0$ for every nonzero $x\in {ℝ}^{n\times 1}$.

• All eigenvalues? of $A$ are positive.

• $A=B^{\top }B$ for some nonsingular? matrix $B$.

• $A$ has an LU decomposition? where all pivots are positive.

• The leading principal minors of $A$ are positive.

• All principal minors of $A$ are positive.

References

• Meyer, Carl D. (2000): Matrix Analysis and Applied Linear Algebra. Philadelphia: SIAM.