Positive-definite matrix

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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[change | change source]

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[change | change source]

  • 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.