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.
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.