Matrices and Vectors

R symbol

all real numbers, also known as scalar

E symbol

denotes membership

scalar

a single numerical quantity, also known as real number

vector in Rn

list of n scalars organized vertically into a list

coordinates

scalars in a vector

m x n matrix

matrix with m rows and n columns

transposition

interchanging the rows and columns of a matrix

linear operation

a mathematical function that preserves addition and scaling

involution

an operator that if applied twice it returns the matrix to the original

symmetric

matrix whose transpose is the same as the original

square

matrix that has the same number of rows and columns

trace

sum of the diagonal

upper triangular

every entry below the diagonal is equal to zero

lower triangular

every entry above the diagonal is equal to zero

diagonal matrix

if all non-diagonal entries are zero

zero matrix

matrix with all zeroes

nonzero matrix

matrix with at least one nonzero entry

nxn identity matrix

matrix with ones on the diagonal and zeros everywhere else

first pivot column

first nonzero column of a matrix

rank one matrix

every column of the matrix is a scalar multiple of the first column

weights

scalars in a linear combination

Inv = v

identity rule

A(c1*v1+c2*v2) = c1Av1 +c2Av2

linearity rule for matrices

zero rule

AOn = Om

eigenvector

Av = lambda*v

eigenvalue

lambda in Av = lambda*v

Xa(lambda) = lambda(In)-A

characteristic equation of a square matrix

directed digraphs

nodes and arrows

number of nodes-number of arrows

Euler characteristic of digraph

connected component

island in a digraph

path

a sequence of distinct arrows joining consecutive nodes

cycle

path that starts and ends at the same node

circuit rank

minimum number of arrows that must be removed to break all cycles

weights in - weights out

net flow

incidence matrix

matrix that encodes all the data in a digraph

path vector

encodes data of a path

cycle vector

encodes data of a cycle

sq.rt. v1² + v2² +…+vn²

length of vector

<v,w>

inner product of v and w

<Av,w> = <v,ATw>

adjoint formula

square length

inner product of a vector with itself

|<v,w>| </= ||v||*||w||

Cauchy-Schwarz inequality

<v,w> = ||v||*||w||*costheta

how to determine inner product with single vector values

acute

if inner product of two vectors is greater than zero

obtuse

if inner product of two vectors is less than 0

orthogonal

if inner product of two vectors is 0

(AB)T = (BTAT)

transposition rule for matrix multiplication

ATA

Gramian

rank

number of columns that have pivots

nullity

number of columns that do not have pivots

rank plus nullity equals number of columns

rank-nullity theorem

linear combinations

non-pivot columns in rref are _________ of the pivot columns

solution vector

vector whose coordinates solve each equation

consistent

when a system has at least one solution

pivot in the augmented column

the system is inconsistent if there is a __________

at least one free variable

the system has infinite solutions if there is ________

row switching, row scaling, row addition

three elementary row operations

rank(A) = rank(AT)

Rank-Transpose theorem

rank(A) = rank(ATA)

Rank-Gramian Theorem

full column rank

all the columns have pivots

full row rank

every row has a pivot

rank deficient

the number of columns/rows is more than the number of pivots

rref(A)

the matrix A has the same column relations as ______

nonsingular

number of rows equals number of columns equals number of pivots

singular

number of rows equals number of columns which is greater than the number of pivots

rref(A)

the matrix A has the same column relations as _____

number of solutions

a matrix system has the same _______ as its reduced row echelon form version

identity matrix

reduced row echelon form of any nonsingular matrix

invertible

if a matrix A has both a left and right inverse

invertible

if A is square and it you can find a left or right inverse, A is ______.

number of shared vertices, direction of arrow

for an incidence matrix, what does <ai, aj> tell us

2

diagonal values of gramian of incidence matrix are always

x = 0

only solution to Ax = 0 for a nonsingular matrix

augment A with Im and reduce until A is the identity matrix

how to calculate inverse

augment A with Im and reduce until A is rref(A)

how to calculate e for ea = r

orthogonal

if PT = P inverse