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f : X -> Y
g : Y -> X
if inverse exists, g = f^{-1}
g(f(x)) = x for all x \in X
f(g(x)) = y for all y \in Y
For simplicity's sake, we will require bijectivity to define the
inverse, although degenerate cases (i.e. non-injective) can be defined.
Matrix Inverse
A := mxn matrix.
Ax where a is nx1 matrix. A can be considered as a function from R^n to R^m.
Definition:
nxn matrix A is invertible iff there exists B nxn such that AB = BA =
I_n. A^{-1} := B.
Thm: If A, B are inverses s.t. AB = I_n, BA = I_n.
A = [ a1 | a2 | ... an ]
B = [ b1 | b2 | ... bn ]
AB = [Ab1 | Ab2 | ... Abn ]
Let e_i = [ 0 0 ... 1 ... 0 ] where 1 is in the ith position.
This gives systems Ab1 = e1, Ab2 = e2 ...
Each can be solved like a standard augmented matrix.
However, we can solve like
[A | e1 | e2 | e3 ... ] (*)
Two possibilities:
- n pivots (every column has pivot)
Reduced echelon form is I_n
Right matrix = B = A^{-1}
- <n pivots (implies at least one row of zeroes at the bottom)
The right matrix is always invertible [how?], so at least one of the
systems in (*) has no solution, and A is not invertible.
If we only use A_j + cA_i -> A_j where j > i to solve
[ A | I_n ],
we get [ U | L^{-1} ]
U is invertible <=> all diagonal elements of U are non-zero
<=> every column of U has a pivot column
L is always invertible, so iff U is invertible, A = LU is invertible.
Transpose
A := mxn matrix.
A^T = B
B := nxm where b_ji = a_ij
A : R^n -> R^m
B : R^m -> R^n (Not inverse properties)
If A is invertible, then A^T is invertible, and
(A^{-1})^T = (A^T)^{-1}
But why?
(1)
If A, B are invertible, AB is invertible, and:
(AB)^{-1} = B^{-1}A^{-1} [why??] [this should verify the previous
identity]
(2)
(AB)^T = B^T A^T [could be proved by brute calculation]
Definition: nxn matrix A is symmetric if A = A^T
If A is symmetric and invertible, A = LU = LDL^T (Thm!)
Then, D would be invertible. If A not invertible, U not invertible, and
D doesn't need to be invertible.
This is Cholesky decomposition. "Keeps the symmetry" (?)
D is a diagonal (and therefore symmetric) matrix.
Chapter 2
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Vector space is a collection V of objects called vectors, a binary
addition operator, and an operator to multiply a vector and a scalar
(defined in R or C)
(u + v) + w = u + (v + w)
a(u + v) = au + av
+ Some more rules (probably commutative?)
(a+b)u = au + bu. Gives existence of 0 vector.
+, * must be closed under V.
Ex: Let V = polynomials degree <= 2.
Ex: Upper-diagonal 2x2 matrices
Ex: R^2
Ex: Subspace of R^2
Not ex: Line in R^2 not containing origin.
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