Vocabulary/percentdot
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%. y Matrix Inverse
Rank 2 -- operates on tables of y -- WHY IS THIS IMPORTANT?
- The inverse matrix of square matrix y, denoted y^-1^
- If y is not square, %. y is the left inverse of y, defined below.
]P=: 0 0 1 , 1 0 0 ,: 0 1 0 NB. permutation matrix 0 0 1 1 0 0 0 1 0 %. P NB. The inverse of P is also a permutation matrix 0 1 0 0 0 1 1 0 0
Common Uses
1. To reverse the effect of a linear transformation.
2. Given a rotation matrix, to form the matrix that rotates in the opposite direction.
sin=: 1&o.
cos=: 2&o.
angle=: (o.2) % 12 NB. 30 degrees, in radians
] R=: 2 2$ (cos,-@sin,sin,cos) angle NB. R: rotate by (angle)
0.866025 _0.5
0.5 0.866025
] S=: 2 2$ (cos,-@sin,sin,cos) -angle NB. S: rotate by (-angle)
0.866025 0.5
_0.5 0.866025
S -: %.R NB. S is the inverse of R
1
R -: %.S NB. R is the inverse of S
1
3. If y is a vector, %. y is a vector in the same direction as y but with reciprocal length.
Related Primitives
Matrix Product (x +/ . * y)
More Information
1. If the columns of y are dependent, domain error is signaled.
2. If y is not square, %. y is (in standard mathematical notation) ((y*y)^-1^y*) where y* is the adjoint of y, that is the complex conjugate of the transpose of y (for real matrices, this is the same as the transpose of y).
With this definition it is always true that (%. y) +/ . * y is an identity matrix.
Details
1. If y is an atom or a list, it is reshaped to a 1-column table, and the result is reshaped to the same shape as y.
1. If y is a table, the shape of %. y is the reverse of the shape of y; put another way, %. y has the same shape as |: y.
x %. y Matrix Divide
Rank _ 2 -- operates on the entirety of x and tables of y -- WHY IS THIS IMPORTANT?
x %. y is (%. y) mux x where mux denotes matrix multiplication.
- When y is square, the result R of x %. y solves the matrix equation y mux R -: x.
-: corresponds to = in standard mathematical notation.
- When y is not square, the result R comes as close as possible to solving the equation, in the minimum-squared-error sense.
sin=: 1&o.
cos=: 2&o.
mux=: +/ . * NB. matrix-multiply
angle=: (o.2) % 12 NB. 30 degrees
]R=: 2 2$ (cos,-@sin,sin,cos) angle NB. rotate by (angle=30 deg)
0.866025 _0.5
0.5 0.866025
angle=: (o.2) % 24 NB. 15 degrees
]Q=: 2 2$ (cos,-@sin,sin,cos) angle NB. rotate by (angle=15 deg)
0.965926 _0.258819
0.258819 0.965926
angle=: (o.2) % 8 NB. 45 degrees
]P=: 2 2$ (cos,-@sin,sin,cos) angle NB. rotate by (angle=45 deg)
0.707107 _0.707107
0.707107 0.707107
Q mux R NB. Rotate successively by 15 deg, then by 30 deg (total=45 deg)
0.707107 _0.707107
0.707107 0.707107
P -: Q mux R NB. Rotation by 45 degrees is the same even if done in 2 steps
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(P %. R) -: Q NB. Matrix-divide both sides of the equation by R
1
Common Uses
1. To solve a system of linear equations.
2. To calculate a linear regression:
indep =. 1 2 3 5 6 NB. independent variable dep =. 3 4 5 8 9 NB. dependent variable dep %. (indep ,. 1) NB. calculate line of best fit 1.24419 1.56977
The best fit is dep = 1.57 + 1.24 * indep.
3. To calculate the current flows in a given electrical circuit network using Kirchhoff's Laws.
Related Primitives
Matrix Product (x +/ . * y)
More Information
1. The columns of y must be independent, or domain error is signaled.
1. In standard mathematical notation, the result is ((y*y)-1y*x) where y* is the adjoint of y, i.e. the complex conjugate of the transpose of y (for real matrices, this is the same as the transpose of y). If y is square this simplifies to (y-1x)
1. If y is an atom or a list, it is reshaped to a 1-column table.
1. If x is a list, it is reshaped to a 1-column table.
1. If x is an atom, it is repeated as needed to make a 1-column table that conforms to %. y. 1. The shape of x %. y is y ,&}.&$ x, which is rather subtle. If y and x are both tables, the result has many rows as y has columns, and the same number of columns as x. But if x or y is an atom or list, the corresponding axis of the result (which would be of length 1) is omitted. Thus:
$ (,. 1 2 3) %. (,. 6 7 9) NB. 3x1 table %, 3x1 table: 1x1 result 1 1 $ (1 2 3) %. (,. 6 7 9) NB. list %. 3x1 table: result is 1-atom list 1 $ (1 2 3) %. (6 7 9) NB. list %. list: scalar result.