square root of positive definite matrix

Suppose M is a positive definitePlanetmathPlanetmath Hermitian matrixMathworldPlanetmath. Then M has a diagonalization


where P is a unitary matrixMathworldPlanetmath and λ1,,λn are the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of M, which are all positive.

We can now define the square root of M as the matrix


The following properties are clear

  1. 1.


  2. 2.

    M1/2 is Hermitian and positive definite.

  3. 3.

    M1/2 and M commute

  4. 4.


  5. 5.

    (M1/2)-1=(M-1)1/2, so one can write M-1/2

  6. 6.

    If the eigenvalues of M are (λ1,,λn), then the eigenvalues of M1/2 are (λ1,,λn).

Title square root of positive definite matrix
Canonical name SquareRootOfPositiveDefiniteMatrix
Date of creation 2013-03-22 15:16:42
Last modified on 2013-03-22 15:16:42
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 12
Author rspuzio (6075)
Entry type Definition
Classification msc 15A48
Related topic CholeskyDecomposition