Essays/Minors
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A minor of size one less than the size of a matrix obtains by removing row i and column j ; the array of such minors can be computed using the outfix (\.):
] m=: i.3 4 0 1 2 3 4 5 6 7 8 9 10 11 minors=: 1&(|:\.)"2^:2 $ minors m 3 4 2 3 <"_2 minors m ┌───────┬───────┬──────┬──────┐ │5 6 7│4 6 7│4 5 7│4 5 6│ │9 10 11│8 10 11│8 9 11│8 9 10│ ├───────┼───────┼──────┼──────┤ │1 2 3│0 2 3│0 1 3│0 1 2│ │9 10 11│8 10 11│8 9 11│8 9 10│ ├───────┼───────┼──────┼──────┤ │1 2 3 │0 2 3 │0 1 3 │0 1 2 │ │5 6 7 │4 6 7 │4 5 7 │4 5 6 │ └───────┴───────┴──────┴──────┘
Complementary indexing also lends itself to the computation of minors:
] i=: <&.>&.> {i.&.>$m ┌─────────┬─────────┬─────────┬─────────┐ │┌───┬───┐│┌───┬───┐│┌───┬───┐│┌───┬───┐│ ││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐││ │││0│││0│││││0│││1│││││0│││2│││││0│││3│││ ││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘││ │└───┴───┘│└───┴───┘│└───┴───┘│└───┴───┘│ ├─────────┼─────────┼─────────┼─────────┤ │┌───┬───┐│┌───┬───┐│┌───┬───┐│┌───┬───┐│ ││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐││ │││1│││0│││││1│││1│││││1│││2│││││1│││3│││ ││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘││ │└───┴───┘│└───┴───┘│└───┴───┘│└───┴───┘│ ├─────────┼─────────┼─────────┼─────────┤ │┌───┬───┐│┌───┬───┐│┌───┬───┐│┌───┬───┐│ ││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐││ │││2│││0│││││2│││1│││││2│││2│││││2│││3│││ ││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘││ │└───┴───┘│└───┴───┘│└───┴───┘│└───┴───┘│ └─────────┴─────────┴─────────┴─────────┘ $ i{m 3 4 2 3 <"2 i{m ┌───────┬───────┬──────┬──────┐ │5 6 7│4 6 7│4 5 7│4 5 6│ │9 10 11│8 10 11│8 9 11│8 9 10│ ├───────┼───────┼──────┼──────┤ │1 2 3│0 2 3│0 1 3│0 1 2│ │9 10 11│8 10 11│8 9 11│8 9 10│ ├───────┼───────┼──────┼──────┤ │1 2 3 │0 2 3 │0 1 3 │0 1 2 │ │5 6 7 │4 6 7 │4 5 7 │4 5 6 │ └───────┴───────┴──────┴──────┘ minors1=: <&.>&.>@{@(i.&.>"_)@$ { ] (minors -: minors1) m 1
Laplace expansion of the determinant of a square matrix:
det=: -/ .* ] m=: _8 + 5 5 ?@$ 20 2 _1 10 1 _6 4 7 _3 7 5 10 2 _8 _3 1 _2 6 _8 _2 _4 1 0 7 9 _4 ] cofactors=: (_1^+/&>{i.&.>$m) * det minors m _2560 _8740 _7780 6540 460 _640 _8510 _3870 810 _5110 _5760 3490 3010 _630 2410 2240 _4425 2215 355 5235 _640 7770 5370 _9310 5890 det m _70400 i=: ?#m +/ (i{m) * i{cofactors _70400 j=: ?{:$m +/ (j{"1 m) * j{"1 cofactors _70400 +/ m * cofactors _70400 _70400 _70400 _70400 _70400 +/"1 m * cofactors _70400 _70400 _70400 _70400 _70400
See also
- Cholesky Decomposition
- LU Decomposition
- QR Decomposition
- Matrix Inverse
- Triangular Matrix Inverse
- Determinant
- Minors
- Hilbert Matrix
- Block Matrix Inverse
- Kronecker Product
Contributed by Roger Hui. The two phrases for minors appear in the Special Matrices section of J Phrases; the first phrase can be found on the green J mug.