Definitions

Remeber it's row x column!
[A] = [a_row,col] = [a_ij] :: nxn Square Matrix A: row i, column j.
- order(A) = n; r = rank(A) ≤ order(A) ≡ r ≤ n.
- [A] is nonsingular when r = n.
{x} :: nx1 Column Vector x, elements in rows {x_j}.
<x> = {x}^T :: 1xn Row Vector x, elements in columns <x_i>.

[A]{x} = {y} = $[a_{ik}][x_{k}] = lbrace y_i rbrace = lbrace { sum{k=1}{n}{a_{ik} x_{k}} } rbrace$ .
[A][B] = [P] =[A_{m x n}][B_{n x p}] = [P_{m x p}] = $[a_{ik}][a_{kj}] =[p_{ij}] = [ sum{k=1}{n}{a_{ik} a_{kj}} ]$ .
[A]^T = [a_ji] when [A] = [a_ij].
Symmetric: [A]=[A]^T
Orthogonal: [Q]^T = [Q]^-1 ⇒ Columns are real mutually orthogonal vecotrs.
[A][0] = [0].
[A][I] = [A].
[I] = $[delta_{ij}] : delta_ij = lbrace matrix{2}{1}{ {0~when i ne j,} {1~when i=j.}}$
[M] = [A]^-1 : [A][M] = [M][A] = [A][A]^-1 = [A]^-1[A] = [I] iff [A] is nonsingular.

|u| = √(u,u) :: Vector length.
Scalar product: |u||v|cos θ = ({u},{v})= ({v},{u}) = {u}^T{v} = {v}^T{u} = .
Square of vector length: l^2(u) = {u}^T{u} = ${u_1}^2 + {u_2}^2 + cdots + {u_n}^2$ .

$Gramian({u_1}, cdots, u_m) = delim{|}{matrix{4}{4}{ {{u_1}^2} {{u_1}^T u_2} cdots {{u_1}^T u_m} {{u_2}^T u_1} {{u_2}^2} cdots {{u_2}^T u_m} vdots vdots ddots vdots {{u_m}^T u_1} {{u_m}^T u_2} cdots {{u_m}^2} } }{|} =0$ iff (u) is linearly dependent.

Elementary Operations FIXME

Permutations and Transdormations

[A] ≡ [P_rowop][A] ≡ [A][Q_colop].
[A] ≡ [B] iff [B] = [P][A][Q] : P and Q are nonsingular.
Orthogonal transformation: [Q]^T A Q = [Q]^-1 A Q : [Q]^T = [Q]^-1 ≡ [Q]^T[Q] = [I].
Congruence transformation: [Q]^T A Q.
Similarity transformation: [Q]^-1 A Q.

Properties

	Addition	Multiplication
Associative	[A] + ([B] + [C]) = ([A] +[B]) + [C]	[A]([B][C]) = ([A][B])[C]
Distributive	Nope	[A]([B] + [C]) = [A][B] +[A][C]
Commutative	[A] + [B] = [B] + [A]	Not generally

([A]^T)^T = [A].
([A]^-1)^-1 = [A].
([A][B])^T = [B]^T[A]^T.
|A B| = |A||B|.
|A^-1| = |A|^-1.
|{u}^T{v}| ≤ √{u}^T{u} √{v}^T{v}
$cos theta = {{u^T v}/{sqrt{u^T u} sqrt{v^T v}}}$ (-π < θ ≤ π)

Gauss-Jordan Reduction

FIXME

Real Quadratic Form

$a_11{x_1}^2 +a_22{x_2}^2 + cdots+a_{nn}{x_n}^2 + 2(a_12 x_1 x_2 + a_13 x_1 x_3 + cdots +a_{n-1,n} x_{n-1} x_{n})$

≡ A = <x>[A]{x} = {x}^T[A]{x} = y.

Homogenous when y=0.

= $< matrix{1}{4}{x_1 x_2 cdots x_n} > delim{[}{ matrix{4}{4}{ {a_11} {a_12} cdots {a_1n} {a_21} {a_22} cdots {a_{2n}} vdots vdots ddots vdots {a_{n1}} {a_{n2}} cdots {a_{nn}} } }{]} delim{lbrace}{ matrix{4}{1}{x_1 x_2 cdots x_n} }{rbrace} =y$

= $< matrix{1}{4}{x_1 x_2 cdots x_n} > delim{lbrace}{ matrix{4}{1}{ {a_11 x_1 + a_12 x_2 + cdots + a_{1n} x_n} {a_21 x_1 + a_22 x_2 + cdots + a_{2n} x_n} vdots {a_{n1} x_1 + a_{n2} x_2 + cdots + a_{nn} x_n} } }{rbrace} = y$

= $< matrix{1}{4}{ {(a_11 x_1 + a_21 x_2 + cdots + a_{n1} x_3)} {(a_12 x_1 + a_22 x_2 + cdots + a_{n2} x_3)} {cdots} {(a_{1n} x_n + a_{2n} x_n + cdots + a_{nn} x_n)} } > delim{lbrace}{ matrix{4}{1}{x_1 x_2 cdots x_n} }{rbrace}$ =y

$y_i = {1/2}{{partial A}/{partial x_i}}$ (i=1,2,…,n) ≡ $delim{}{matrix{4}{1}{ {a_11 x_1 + a_12 x_2 + cdots + a_{1n} x_n = y_1,} {a_12 x_1 + a_22 x_2 + cdots + a_{2n} x_n = y_2,} vdots {a_{1n} x_1 + a_{2n} x_2 + cdots + a_{nn} x_n = y_n,} } }{rbrace}$ ≡ [A]{x} = {y}

Characteristic Values

λ_i : [A]{x} = λ_i{x} or ([A] - λ_i[I]){x} = 0.

|A - λI| = 0 = $delim{|}{ matrix{4}{4}{ {(a_11 - lambda)} a_12 cdots a_{1n} a_21 {(a_22 - lambda)} cdots a_{2n} vdots vdots ddots vdots a_{n1} a _{n2} cdots {(a_{nn} - lambda)} } }{|}$ = 0.