block determinants

If A and D are square matricesMathworldPlanetmath

  • If A-1 exists, then

  • If D-1 exists, then


The matrices D-CA-1B and A-BD-1C are called the Schur complements of A and D, respectively.
Mention that

  • If A, D are square matrices, then


    , where O is a zero matrixMathworldPlanetmath.

  • Also we have that

  • Another useful result for block determinants is the following.
    As J=(OI-IO) is a symplectic matrix, we have that detJ=1. Using now the fact that detMN=det(M)det(N) for any M, N square matrices, we have that


This holds for any square matrices A, B and for the last point A, B have also the same order. They do not need to be invertiblePlanetmathPlanetmath.

Title block determinants
Canonical name BlockDeterminants
Date of creation 2013-03-22 15:25:57
Last modified on 2013-03-22 15:25:57
Owner georgiosl (7242)
Last modified by georgiosl (7242)
Numerical id 20
Author georgiosl (7242)
Entry type Theorem
Classification msc 15A15
Related topic SchurComplement
Related topic DeterminantsOfSomeMatricesOfSpecialForm