Symmetric positive-definite matrix

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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.
View other properties of square matrices with entries over the field of real numbers | View other properties of square matrices

Definition

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:

  1. '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 ).
  2. The bilinear form on defined by (where the input vectors are written as column vectors) is a symmetric positive-definite bilinear form.
  3. There is a invertible matrix such that .
  4. is a symmetric matrix and a P-matrix.

Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
invertible matrix |
positive-definite matrix |
P-matrix |