Math 218

square matrix

an m x n matrix is square if m = n

the diagonal of a matrix

refers to the collection of (i,i) entries of the matrix

upper triangular

refers to a matrix if every entry below the diagonal is zero

lower triangular

refers to a matrix if every entry above the diagonal is zero

diagonal

refers to a matrix if every non diagonal entry is zero

diag(d1, .... dn)

is the n x n diagonal matrix w/ the diagonal d1, ... dn (everything else is 0)

zero vector, zero matrix

vector or matrix that are full of zeros. can write O2 or O4x3 for ex. (doesnt have to be square)

nonzero

a matrix or vector if at least one entry is nonzero

In

the n x n identity matrix which has ones on the diagonal and zeros everywhere else

first pivot column

first non-zero column

rank one

every column is a multiple of the first pivot column

vecotr in R^n

is a list of n coordinates organized vertically in a list

matrix m x n

R^(m x n)

scalar v product

vector w. scalar multiplied by each coordinate

vector addition

add up coordinates across the rows

scalar matrix products

multiply scalar by every coordinate in the matrix

matrix addition

add up coordinates with the same position. onlt matrices of the SAME SHAPE can be added togehter

A^T

formed by interchanging rows and columns

transposition is

a linear operation meaning it can be distributed. so, (c1A1 + c2A2)^T = c1At + csAt)

symmetric

sT = S, symmetry across the diagonal

trace(matrix)

the sum of its diagonal, matrix must be square, is a scalar, is a linear operation (can be distributed and can take the scalar out)

linear combination

gives you a vector, in the form: scalar1*v1 + scalar2*v2...+snvn

matrix-vector products

encoded linear combinations, matrix holds vectors and scalars go into v. vector can also be used to add and subtract some of the columns, warning #collumns A must = #rows v!!)

3 matrix-vector multiplication rules

linearity, identity, and zero rules

linearity rule

A(c1v1 + c2v2) = c1Av1 + c2Av2 (matrix x vector, order matters)

identity rule

Inv = v

zero rule

A0n = 0m ,0n is vector, 0m is vector:)

R^n --A--> R^m

sophisticated way of writing A is m x n matrix (not order of rows and columns changes)

eigenvector

v is an eigenvector of A if v != 0, Av = lamdaV, A must be square, --> v∈Ea(lamda)

eigenvalue

lamda, when Av = lamda v

directed graph

collection of nodes and arrows, inlcludes all non-connected parts

digraphs

describe the relationships between objects, for ex: modeling transactions between financial institutions

euler characteristic

of a diagraph G: # of nodes - # of arrows

connected components

"islands" in a diagraph

path

sequence of distinct arrows joining consecutive nodes (only use each arrow one time)

cycle

path that starts and ends at the same node (still only use each arrow once)

circuit rank

the minimum # of arrows that must be removed to break ALL cycles

weights

scalar values added to arrows, (-1, 0, 1) weights convey directionality,

net flow

through a node (phi(v), φ(v) = weights in - weights out. Adding up net flows for each node should be zero, DO include negatives in the calculation

incidence matrix

encodes all of the data in a diagraph, -1 for leaving, 1 for going to, 0 for nothing, has form R^nodes x arrows

weight vector

is a vector that organizes the weights

path vector

encodes the data of the path, will be a vector of -1, 0, and 1's \. when multiplied by incidence matrix --> gives start and end pos (as long as its not a cylce)

cycle vector

of a cycle c is the associated path vector, multiplied by incidence gives all 0s

incidence matrix x weight vector

the netflow of each node! the ith coordinate of Aw is the net flow of the ith node

R^2

is an arrow in the xy-plane, calculate vector w. tail-to-tip displacement = tipx - tailx for top & tipy - taily for bottom, vector reps displacement not location, so same vector could be anywhere on the graph

length of v = [v1 v2]^t

||v|| = sqrt(v1^2 + v2^2), pythag!

v ∈ R^3

an arrow in the x-y-z space

length of v =

sqrt(v1^2 + v2^2 +... +vn^2)

||c*v|| =

|c| * ||v||

the product c*v

scales and orients v, coefficient either makes it longer of shorter & if a neg sign, flip directions

v + w

it means putting them together tail to tip, then form a parallelogram --> the diagonal is the sum

unit vector

vector w/ length one, indicated with v-hat

normalization of v

v-hat = (1/||v||)*v

inner product v, w ∈ R^n =

v1w1 + v2w2 + ... + vnwn, also called dot product but we won't use that here, gives a SCALAR, only wrk if the have the SAME SHAPE

inner product properties

commutative, left-linear, right-linear, adjoint formula, length formula

commutative inner product prop

(

left linear

right linear

adjoint formula

length formula

||v - w||^2

||v||^2 + ||w||^2 -2*

law of cosines

||v-w||^2 = ||v||^2 + ||w||^2 - 2||v|| *||w|| * cos(theta)

||v|| * ||w|| * cos(theta)

corollary (the cauchy-schwarz inequality)

|

acute

obtuse

orthogonal

Matrix mult diagram

R^2ndcol --2nd matrix--> R^2ndrows, 1st col --1st matrix--> R^1strows

multiplying two matrix simplifies

multiplying the A matrix by two vecotrs. put the vecotrs in one matrix and the output matrix will be the two matrix vec products

Row perspective matrix mult

the (i, j) entry of the matrix product AB is given by the inner product ; AB is filled entry by entry w. inner products

Row perspective understanding pos, 0, neg entries

0 entries mean the i row of A is orthogonal to the jth column of B, pos values are acute and neg values are obtuse

matrix arithmetic props

additive associativity, multiplicative associativity, additive commutativity, left distribute, right distribute ,identity rule, trace rule, transportation rule,

transposition

is an order reversing involution (and trace)

additive associativity

A + (B + C) = (A + B) + C

multiplicative associativity

(AB)C = A(BC)

additive commutativity

A + B = B + A

left distributive

A(B + C) = AB + AC

right distributive

(A + B)C = AC + BC

identity rule

AIn = ImA = A

trace rule

trace(AB) = trace(BA) (ONLY SQUARE matrices tho)

transposition rule

(AB)^T = B^TA^T

matrix mult warning

it is NONCOMMUTATIVE, order maters bro

Gramian of A

= A^TA --> square matrix & symmetric, EVERY matrix has a gramian

A^TA also equals

(A^TA)^T, A^T(A^T)^T

row perspective on gramian

each entry (i,j) of A^TA is so

given two vectors, v^Tw = ..

inner product thrm

rref if..

1) any zero-rows occur at the bottom

2) the 2st nonzero entry of the other rows is equal to 1

3) every pivot occurs to the right of the pivots above

4) all nonpivot entries in a column containing a pivot are equal to zero

every matrix can be row

rank(A)

# of pivot cols

non-pivot colms

are linear combinations of the pivot columns (col relations)

row echelon form if

1) any zero-ros occur at the bottom

2) the first nonzero entry of the other rows is called a pivot

3) evry pivot occurs to the right of the pivots above it

4) all entries below a pivot are equal to zero

consistent

a system is consistent if it has at leas one solution,

inconsistent

a system if it has no solutions, if u sum up columns downwards and across and get 0, but augmented/ answers is 1 its inconsistent, IF PIVOT IN THE AUGMENTED COL.

all systems

can be written as an augmented matrix and can be expressed by eq: Ax = b (A is coeff matrix, b is augmented col, x is solution matrix)

depndent

variable associated w/ pivot column/ the pivot column

free

variable associated w. non pivot col/ non-pivot col

rank(A)

of coefficient matrix is # of dep. variables/ pivot cols

rouche-capelli theorem

every rref system falls into one of the following three categories: inconsistent, consistent (one sol, nullity(A) = 0), consistent (infinite sol nullity(A) > 0), so do REF's, back subbing used in ref

solving reduced systems

1) set the free variables to scalars c1.... cn

2) wrtie each dep. variable in terms of c1, c2... cn

3) express the full solution as x = xp + x1x1 + ...+ ckxk

degrees of freedom of a system

nullity(A)