Sylvester's determinant identity
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In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof in 1851.
The identity states that if A and B are matrices of size m × n and n × m respectively, then
The identity may be proved as follows. Let M be a matrix comprising the four blocks Im, −A, B, and In:
Because Im is invertible, the formula for the determinant of a block matrix gives
Because In is invertible, the formula for the determinant of a block matrix gives
- 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.
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