Positive-definite matrix

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.  

 Definition 

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.

 Examples 

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.

Category:Linear algebra

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