Editing Definite matrix (section)

=== Square root ===
{{main|Square root of a matrix}}
A Hermitian matrix <math>M</math> is positive semidefinite if and only if there is a positive semidefinite matrix <math>B</math>  (in particular <math>B</math> is Hermitian, so <math>B^* = B</math>) satisfying <math>M = B B.</math> This matrix <math>B</math> is unique,<ref>{{harvtxt|Horn|Johnson|2013}}, p. 439, Theorem 7.2.6 with <math>k = 2</math></ref> is called the ''non-negative [[square root of a matrix|square root]]'' of <math>M,</math> and is denoted with <math>B = M^\frac{1}{2}.</math>
When <math>M</math> is positive definite, so is <math>M^\frac{1}{2},</math> hence it is also called the ''positive square root'' of <math>M .</math>

The non-negative square root should not be confused with other decompositions <math>M = B^* B.</math>
Some authors use the name ''square root'' and <math>M^\frac{1}{2}</math> for any such decomposition, or specifically for the [[Cholesky decomposition]],
or any decomposition of the form <math>M = B B;</math>
others only use it for the non-negative square root.

If <math>M \succ N \succ 0</math> then <math>M^\frac{1}{2} \succ N^\frac{1}{2} \succ 0.</math>