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Determinants of Special Matrices

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For some special types of $ n\times n$-matrices $ A$ the determinant can be given immediately.

(i)
Triangular matrices: If $ a_{i,j}=0$ for $ i<j$ or $ i>j$, then we can calculate the determinant by

$\displaystyle \operatorname{det}A = a_{1,1}\cdots a_{n,n}\, . $

(ii)
Block-diagonal matrices: If matrix $ B$ has a blocked structure with $ A_{i,j}=0$, $ i\neq j$ and square diagonal blocks, then we have

$\displaystyle \operatorname{det}B = \prod\limits_{i=1}^k \operatorname{det}A_{i,i}\,. $

(iii)
For a unitary matrix $ U$ we have

$\displaystyle \vert\operatorname{det}U\vert=1\,. $

In particular, for orthogonal matrices $ (u_{i,j}\in \mathbb{R})$ we have

$\displaystyle \operatorname{det}U \in \{-1,1\}\,. $

(Authors: Burkhardt/Höllig/Hörner)

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[Annotations]
  automatically generated 2/ 9/2005