Weak positive definite matrix

A weakly positive definite matrix is a real matrix whose real eigenvalues ​​are all positive and whose eigenvectors form a generating system. One can define these matrices equivalently by saying that the matrix has real entries and one can decompose them into , where is any invertible matrix and has positive or non-real diagonal elements. ${\displaystyle W}$ ${\displaystyle W=XDX^{-1}}$ ${\displaystyle X}$ ${\displaystyle D}$

From this one derives a property that weakly positive definite matrices can always be written as the product of two positive definite matrices . Some authors also define matrices that are weakly positive. This shows that every positive definite matrix is ​​also a weakly positive definite matrix. For one, any positive definite matrix with the unitary matrix multiply and receives again the (weak) positive definite matrix: . ${\displaystyle A}$ ${\displaystyle I}$ ${\displaystyle AI=A}$

Weak positive definite matrices are used in solving time-dependent partial differential equations with the help of the Runge-Kutta scheme .

Individual evidence

1. ↑ Eugene Paul Wigner : On Weakly Positive Matrices . In: The Collected Works of Eugene Paul Wigner . S. 559-563 , doi : 10.1007 / 978-3-662-02781-3_40 .
2. ↑ ^{a b} T. K. Nilssen: Weakly positive definite matrices. (PDF) Accessed February 8, 2018 .