matrix
A[e1,e2,...,eN]
transformation
column i are image of standard basis ei
space(cols or rows)
a box
formed by columns or rows as edges
Each standard basis vector Ei is transformed to A column Ci
Av = AIv
one basis to another
column i are image of standard basis ei
orthogonal matrix vs unitary matrix
Projection Matrix
e = b - p
p = xa
a^t e = 0
a^t (b - xa) = 0
x = a^t b / a^t a
p = xa
Proj-p = Pb
Pb = xa = (aTb/aTa)a
= (aaT/aTa)b
P = (aaT/aTa) = aaT/||a||
P = uuT (u = a/||a||)
Ax = b
b not in colspace(Ax) -> no solution
closest vector to b is b's projection on colspace(Ax)
get x when Ax = b-proj onto colspace(Ax)
b's projection onto a space is a vector
in the space which is closest to b
Inverse & Transpose
Rank
rank(A+B)≤rank(A)+rank(B)
rank(A+B)≥max(rank(A),rank(B))−min(rank(A),rank(B))
A MxN, B MxN
rank(A+B) <= min(M, N, rank(A)+rank(B))
Subset of rank one matrices is not subspace
b = p + e
b = Pb + e
e = b - Pb = (I-P)b
Pe = (I-Pp)
Pp = (I-Pe)