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Introduction to Vectors and Matrices Introduction to Vectors and Matrices

Introduction to Vectors and Matrices - PowerPoint Presentation

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Introduction to Vectors and Matrices - PPT Presentation

Matrices Definition A matrix is a rectangular array of numbers or symbolic elements In many applications the rows of a matrix will represent individuals cases people items plants animals and columns will represent attributes or characteristics ID: 547960

vectors matrix columns rows matrix vectors rows columns matrices rank linear form number linearly inverse dimensional space examples multiplication

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Slide1

Introduction to Vectors and MatricesSlide2

Matrices

Definition: A matrix is a rectangular array of numbers or symbolic elements

In many applications, the rows of a matrix will represent individuals cases (people, items, plants, animals,...) and columns will represent attributes or characteristicsThe dimension of a matrix is its number of rows and columns, often denoted as r x c (r rows by c columns)Can be represented in full form or abbreviated form:Slide3

Special Types of MatricesSlide4

Regression Examples - Carpet DataSlide5

Matrix Addition and SubtractionSlide6

Matrix MultiplicationSlide7

Matrix Multiplication Examples - ISlide8

Matrix Multiplication Examples - IISlide9

Special Matrix TypesSlide10

Linear Dependence and Rank of a Matrix

Linear Dependence: When a linear function of the columns (rows) of a matrix produces a zero vector (one or more columns (rows) can be written as linear function of the other columns (rows))

Rank of a matrix: Number of linearly independent columns (rows) of the matrix. Rank cannot exceed the minimum of the number of rows or columns of the matrix. rank(A) ≤ min(rA,ca) A matrix if full rank if rank(A) = min(rA,

ca) Slide11

Geometry of Vectors

A vector of order n is a point in n-dimensional space

The line running through the origin and the point represented by the vector defines a 1-dimensional subspace of the n-dim spaceAny p linearly independent vectors of order n, p < n define a p-dimensional subspace of the n-dim spaceAny p+1 vectors in a p-dim subspace must have a linear dependencyTwo vectors x and y are orthogonal if x’y = y’

x = 0 and form a 90 angle at the originTwo vectors x and

y

are linearly dependent if they form a 0

 or 18

0

 angle at the origin

Slide12

Geometry of Vectors - II

If two vectors each have mean 0 among their elements then

q

is the product moment correlation between the two vectorsSlide13
Slide14

Matrix Inverse

Note: For scalars (except 0), when we multiply a number, by its reciprocal, we get 1: 2(1/2)=1

x(1/x)=x(x-1)=1In matrix form if A is a square matrix and full rank (all rows and columns are linearly independent), then A has an inverse: A-1 such that:

A-1 A = A A-1

=

ISlide15

Computing an Inverse of 2x2 MatrixSlide16

Use of Inverse Matrix – Solving Simultaneous EquationsSlide17

Useful Matrix ResultsSlide18

Orthogonal MatricesSlide19

Eigenvalues and Eigenvectors Slide20

Positive Definite Matrices / Spectral DecompositionSlide21

Distance as a Quadratic FormSlide22
Slide23

Square Root of a Positive Definite Square MatrixSlide24

Mean, Variance, Covariance, CorrelationSlide25

Random Vectors and MatricesSlide26

Mean and Variance of Linear Functions of XSlide27

Standard Deviation and Correlation Matrices

LPGA “Population” Data:Slide28

Partitioned Covariance MatrixSlide29

Matrix Inequalities and MaximizationSlide30

Multivariate Normal Distribution