11 - Bases of new vector spaces, Rank one matrices

All for 3x3 cases

Basis - 9 dimensional (1 at each cell)
Basis of symmetric matrices - 6 dimensional
Basis of upper triangular matrices - 6 dimensional
Basis of (Symmetric ∩ Upper triangular) - 3 dimensional (since diagonal)
Basis of (Symmetric ∪ Upper triangular) - 9 dimensional (since all 3x3)
Every rank 1 matrix can be expressed as - u x v^T
- where u and v are column matrices
Let M be all 5x17 matrices with rank 4. Is M a subspace?
- No. Since no 0 matrix

12 - Graphs and Networks, Incidence matrices, Kirchoff’s Law

Consider this directed graph -

Here m = 5 edges, n = 4 nodes
A - Incidence matrix can be used to denote this graph
Let Null space solution of A be x -
Ax denotes the Potential Difference between nodes
- shows circuit movement only when potential difference
Let C be a matrix which takes us from Potential Difference to Current through edges (Ohm’s law)
C - Conductance matrix
Let current be Y -
A^Ty = 0 - Solving Left null space (Kirchoff’s current law)
- Solving left null space just means current coming IN a node is equal to current going OUT through it
- can find basis by assuming current (y) through one edge and solving such that no charge accumulation
- Dimension of Left Null Space = Number of independent loops
- Rank = Number of nodes - 1
- dim(N(A^T)) = m - r
- #loops = #edges - (#nodes - 1)
- #nodes - #edges + #loops = 1 –> Euler's Formula

x and y are orthogonal if - x^Ty = 0
Two subspaces are orthogonal if every vector in subspace 1 is perpendicular to every other vector in subspace 2
row space is perpendicular to null space
column space is perpendicular to left null space

Why projections?
- Because Ax = b may not have a solution
- projection means changing b to closest vector in column space of A
- Ax^{^} = p

[Diagram]

a ⊥ e
a^Te = 0
a^T(b - p) = 0
a^T(b - xa) = 0
x.a^T.a = a^Tb
x = (a^Tb) / (a^Ta) - scalar
p = ax = a ((a^Tb) / (a^Ta)) - vector (closest b)
Since projection of b, p is governed by b i.e scaling b also scales p
So we can write p as:
- p = (a a^T / (a^Ta)) b
- here (a a^T / (a^Ta)) is known as Projection matrix (P)
Column space of P - line a
rank(P) = 1
P = P^T …. symmetrical
P² = P …. projecting same line twice, thrice will not change projection

[Diagram]

a₁^Te = 0 = a₂^Te
e = b - p = b - Ax^{^}
a₁^T(b - Ax^{^}) = 0 = a₂^T(b - Ax^{^})
Combining both equations (writing them in matrix form)
- A^T(b - Ax^{^}) = 0
- A^TAx^{^} = A^Tb
x = (A^TA)^-1A^Tb
p = A(A^TA)^-1A^Tb
Here projection matrix P is A(A^TA)^-1A^T
If A is square - P = I …. since column space is whole space
P = P^T …. symmetrical
P² = P …. projecting same line twice, thrice will not change projection