Sylvester's determinant theorem
In matrix theory, Sylvester's determinant theorem is a theorem useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this theorem without proof in 1851.
The theorem states that if A, B are matrices of size p × n and n × p respectively, then
This can be seen for invertible A, B by conjugating I + AB by A-1, then extended to arbitrary square matrices by density of invertible matrices, and then to arbitrary rectangular matrices by adding zero column or row vectors as necessary.
|This section does not cite any references or sources. (December 2014)|
The theorem may be proven as follows. Let be a matrix comprising the four blocks , , and
Block LU decomposition of yields
follows. Decomposing to an upper and a lower triangular matrix instead,
- Sylvester, James Joseph (1851). "On the relation between the minor determinants of linearly equivalent quadratic functions". Philosophical Magazine 1: 295–305.
Cited in Akritas, A. G.; Akritas, E. K.; Malaschonok, G. I. (1996). "Various proofs of Sylvester's (determinant) identity". Mathematics and Computers in Simulation 42 (4–6): 585. doi:10.1016/S0378-4754(96)00035-3.
- Harville, David A. (2008). Matrix algebra from a statistician's perspective. Berlin: Springer. ISBN 0-387-78356-3. page 416
- Weisstein, Eric W. "Sylvester's Determinant Identity". MathWorld--A Wolfram Web Resource. Retrieved 2012-03-03.