Symmetric positive-definite matrix
This article defines a property that can be evaluated for a square matrix with entries over the field of real numbers. In other words, given a square matrix (a matrix with an equal number of rows and columns) with entries over the field of real numbers, the matrix either satisfies or does not satisfy the property.
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Suppose is a positive integer and is a square matrix. We say that is a symmetric positive-definite matrix if the following equivalent conditions hold:
- 'Symmetric and positive-definite: (i.e., is a symmetric matrix: it equals its matrix transpose) and is a positive-definite matrix: for every column vector , we have that , and equality holds if and only if is the zero vector (in other words, for all nonzero column vectors ).
- The bilinear form on defined by (where the input vectors are written as column vectors) is a symmetric positive-definite bilinear form.
- There is a invertible matrix such that .
- is a symmetric matrix and a P-matrix.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|