Positive-definite matrix

algebra concept

A positive-definite matrix is a matrix with special properties. The definition of the term is best understood for square matrices that are symmetrical, also known as Hermitian matrices.


A square matrix filled with real numbers is positive definite if it can be multiplied by any non-zero vector and its transpose and be greater than zero. The vector chosen must be filled with real numbers.


  • The matrix   is positive definite. To prove this, we choose a vector with entries  . When we multiply the vector, its transpose, and the matrix, we get:  

when the entries z0, z1 are real and at least one of them nonzero, this is positive. This proves that the matrix   is positive-definite.