## Matrices

 LA home Computing Maths  Glossary  Calculus  Stats  Series  N χ  Trig  Vectors  Matrices  Eigen v.  Ortho.  Least squares  Quaternions  Lagrange mult.  Matrices also see algorithms  numerical   matrices
f is a linear-function (-map, -transformation, ...) iff
f(v + w) = f(v) + f(w),
f(k v) = k f(v)
(consequently
f(0) = 0.)
f is often called a linear-transformation if the input and output spaces of f are the same.
A linear transformation over a finite-dimension vector-space can be represented by a matrix.

#### Examples

Identity, I:
I =
 1 0 0 1
I
 x y
=
 x y

Projection of <x, y>, onto the line through the origin whose unit normal is n = <nx, ny>:
 1-nx2 -nxny -nxny 1-ny2
 x y
(<x, y> → <x, y> - (<x, y> . n) n = <x, y> - (x nx - y ny) n = <x - (x nx + y ny) nx, y - (x nx + y ny) ny>.)

Reflection of <x, y>, in the line through the origin whose unit normal is n = <nx, ny>:
 1-2nx2 -2nxny -2nxny 1-2ny2
 x y

Anti-clockwise rotation of <x, y>, by angle θ, about the origin:
 cosθ - sinθ sinθ cosθ
 x y

#### Equations

Given constants a, b, c, d, p, q, s, and t, and variables w, x, y, and z, in matrices
 a b c d
 w x y z
=
 p q r s
i.e.,
a w + b y = p,
a x + b z = q,
c w + d y = r,
c x + d z = s.

The following are all 2×2 but generalize ...

 a b c d
 w x y z
=
 p q r s
is equivalent
(wrt solving for
w, x, y, and z) to
 b a d c
 y z w x
=
 p q r s
col
swap
row
swap

to
 c d a b
 w x y z
=
 r s p q
row
swap
row
swap

to and to
 ka kb c d
 w x y z
=
 kp kq r s
row multiplication
by constant k
 a+kc b+kd c d
 w x y z
=
 p+kr q+ks r s
(or subtraction)

If we can work on M and P to reduce
M X = P
to an equivalent
I X = P',
using the relations above, then we can just read off the solution, P', for X.
X and P can be n×1 column vectors, or n×n matrices, etc..
For example, let P=I, the identity, and reduce
M X = I
to
I X = M-1
giving the matrix inverse of M.
(Note that any column swaps cause row swaps, in X, which must be undone to get the final answer.)