Matrices Practice

8/13/2019 Matrices Practice

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Matrices

COMM2M

Harry R. Erwin, PhDUniversity of Sunderland


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Basic Concepts

Vector space

Linear transformation


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Resources

Korevaar, J., 1968,Mathematical Methods,

Academic Press.


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Vector Space

A vector space or linear space Vis a collection of

elements with the following properties:

Ifxandy are any two elements of V then V contains anelement that may be called the sumx + y ofx andy.

Ifx is any element of V and an arbitrary scalar

(element of the underlying commutative field) then V

also contains an element that can be called the scalarmultiple x.

In applied mathematics, the underlying commutative

field is almost always the real or complex numbers.


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Vector Space Axioms1. Existence of vector addition.

2. Addition is commutative.

3. Addition is associative.

4. Existence of a unique zero vector.

5. Existence of a unique additive inverse.

6. Existence of scalar multiplication.

7. Scalar multiplication is associative.8. Multiplication by 1 is the identity.

9. Multiplication by 0 produces the zero vector.

10. Multiplication and addition interact as expected.


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Examples of Vector Spaces

Ordinary n-dimensional space,Rn.

Complex n-dimensional space, Cn.

Infinite sequences of real or complex numbers.

The continuous functions, C(a,b),defined on a finiteinterval (a,b).

Cn(a,b),the continuous functions on the same

interval with ncontinuous derivatives. The integrable functions on the same interval with a

linear norm, 1[a,b] or squared norm, 2[a,b].

The linear maps betweenRnandRm(or Cnand Cm).


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Linear Transformation

A transformation from one vector space intoanother (or into itself) that commutes withaddition and scalar multiplication. If such atransformation is denotedL:

L(x+y) = Lx + Ly

Lx = Lx

Ordinary differentiation is an example. Everylinear ordinary or partial differential operator is alinear transformation.

Laplace, Fourier, and other integral

transformations are linear.


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Matrices and Linear

Transformations Matrix productsAxT(whereAis a matrix,

andxTa column vector) can define linear

transformations with respect to given finitedimensional bases over a commutative

field.

Many linear problems in engineering andmathematics can be written in the form:

Lx = z.


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Vector Spaces and Coordinate

Bases A basis of a vector space is a collection of

vectors so that any vector in the space can

be uniquely described as a linearcombination of those basis vectors.

Every vector space has such a basis.

The number of such basis vectors is thedimension of the vector space. This can beshown unique.


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Scalar Norms

For a real number, r,the absolute value of

the number, |r|,is defined to be r if r > 0,

and otherwise -r.

For a complex number, a + bi,the absolute

value of the number is defined to be the

positive square root of (a2+b2).


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Normed Vector Spaces

A norm for a vector space is a map, m,fromthe vector space into the non-negative real

numbers that has the following properties:m(v) = ||m(v)

m(a+b) m(a) + m(b)

m(x) = 0 iff x is the zero vector

mcan be referred to as a distance functionand written ||x||.


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Properties of Normed Vector

Spaces In a normed vector space, every basis vector has a

length > 0. If the underlying scalar field is the

complex or real numbers, each basis vector can bereplaced by a basis vector of length one.

Three useful norms are:

||x||1 = |x1| + . . . + |xn|

||x||2 = (|x1|2+ . . . + |xn|2)

||x||= max(|x1|, . . . , |xn|)


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Inner Product

The inner product of two vectors,xandy, in anormed vector space relative to a specific basis ofunit vectors is defined as:

x.y= xiyi.Wherexiandyi are the coefficients of each vector.

Ifx.y= 0, the vectors are normal to each other.

If bi.bj= 0 for any two different vectors in thebasis, the vector space has an orthonormal basis.This can be very convenient.

A finite dimensional vector space overRor Chas

an infinite number of such bases.


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Matrices and Linear

Transformations Suppose we have a linear transformation,L,

between two finite dimensional vector spaces,FnandFm, each with an identified basis.Lthen can

be written as a nxmmatrixMwith coefficients inF.

If Tis a transformation between two bases ofFn,(T:B

1

-> B2

), then the matrix transformation forLwritten in terms ofB2is T

-1.M. If TtransformsFmtoFm, the matrix transformation isM

.T, (usingmatrix multiplication and inversion).


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Writing Matrices

A matrixAcan this be written as a collection ofelements aij, where i is the row number and j is thecolumn number.

The transpose,AT, is a matrix consisting of aji.

Matrices of the same number of rows and columnscan be summed.

If matrix A has i rows and j columns, and matrix Bhas j rows and k columns, the productA.B isdefined as a matrix C with i rows and k columnssuch that cik= aij

.bjk.


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Other Key Concepts

The rank of a matrix is the rank of thecorresponding linear transformation (thedimension of the space that the lineartransformation maps into). This may besmallerthan the number of columns of the matrix.

Two matricesA, Bare equivalent if they representthe same linear transformation for different bases.

SupposeRand Sare invertibletransformationsbetween bases in the source and destination vectorspaces such thatR.A = B.S. ThenAandBareequivalent.


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Square Matrices

A square matrix represents a linear transformation

from a vector space to itself.

The square matrix,I(defined as aij= 1.0 if i=j and0.0 otherwise)represents the identity

transformation.

An invertible matrixAhas a second unique

matrix,A-1such thatA.A-1= I. HenceArepresents

a one-to-one linear transformation.


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Determinant

The determinant of a linear transformationLfrom avector space to itself is a non-zero function to the scalarfield that computes the (signed) volume of the image of

a n-cube whenL is applied to it. The sign describeswhether the resulting image has the same orientation.

The determinant is independent of the basis used torepresent the transformation.

Algorithms for matrix inversion typically use thedeterminant. Computing eachcomponent of thedeterminant involves nmultiplications, where nis thedimension of the vector space.


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Computational Issues in Matrix

Arithmetic Computer arithmetic is almost always inexact:

When you add two variables, much of the significance

of the smaller variable can be lost. If it is small enoughrelative to the other, it is treated as zero.

When you multiply two variables, the lowest order bit

of the product is noisy.

When you take the difference of two nearly equalnumbers, most of the resulting bits are noise.

Determinants are particularly vulnerable to this.


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Problems in Matrix Arithmetic

Watch out for nearly singular systems of linearequations. These are systems where the determinantis close to zero. Round-off errors are likely to make

these systems linearly dependent. Watch for accumulated round-off errors in systems

with high dimensionality. Your solutions need to betested rather than trusted.

Watch out for systems where the row or columnnorms vary massively. The smaller rows/columnswill lose much of their significance.

MATLAB is designed to handle these problems.


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MATLAB and Matrices

The basic data type in MATLAB is adouble, an array of complex numbers.

At this point, we will consider mby nmatrices. A column vectorhas n = 1and arow vectorhas m = 1.

To access the ijth component of the matrixA, use A(i,j).


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Creating Matrices

zeros(m,n) (or zeros([m,n])) creates an m-by-nmatrix of 0.0.

ones(m,n) creates an m-by-n matrix of 1.0. eye(m,n) creates an m-by-n matrix of 1.0 for i = jand otherwise 0.0.

eye(m) (etc.) creates an m-by-m matrix.

rand(m,n) contains uniformly distributed randomnumbers selected from [0,1].

randn(m,n) contains normally distributed randomnumbers (from the standard normal distribution).


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Literal Matrices

Matrices can be built explicitly using the squarebracket notation:

A = [2 3 57 11 13

17 19 23]

This creates a 3x3 matrix with those values.

Row ends can be specified by ; instead of carriagereturns. Separators are spaces or ,

Dont separate a + or - sign by a space!


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Assembling Matrices

B = [1 2; 3 4]

C = [B zeros(2)

ones(2) eye(2)]

C =

1 2 0 0

3 4 0 01 1 1 0

1 1 0 1


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Other Approaches

Block diagonal matrices can be created by

the blkdiag function.

Tiled matrices can be created by usingrepmat.

There is a large list of special matrices.


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Subscripting and the Colon

Notation The colon is used to define vectors that can act as

subscripts. For integers i and j, i:j is used to denotea row vector from i to j in steps of 1.

A nonunit stride or step is denoted i:s:j.

Matrix subscripts (1 or greater!) are accessed asA(i,j).

A(p:q,r:s) is a submatrix of A. A(:,j) is the jth column and A(i,:) is the ith row.

endrepresents the last column or row.


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Arbitrary Submatrices

A([i j k],[p q]) is the 3x2 submatrix built from the

intersection of the ith, jth, and kth rows with the

pth and qth columns. A(:) is a vector consisting of all the elements of A

taken from the columns in order from first to last.

A(:) = valueswill fill A, preserving its shape.

linspace(a,b,n) will create a vector of n values

equally spaced from a to b. n defaults to 100.


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The Empty Matrix

[] is an empty 0-by-0 matrix.

Assigning [] to a row or column deletes that

row or column from the matrix.

Also is used as a placeholder in argument

lists.


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Matrix and Array Operations

Operation Matrix Array

Addition + +

Subtraction - -

Multiplication * .*

Left Division \ .\

Right Division / ./

Exponentiation ^ .^


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Left and Right Division

a/b means a/b in the usual sense.

a\b means b/a!

For matrices, these are carried out using

matrix operations

A/B means A*B-1 (solving X*B = A)

A\B means A-1*B (solving A*X = B)

For elementwise operations, precede with .


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Matrix Powers

A^n is defined for all powers, including negative

and non-integer.

.^ is elementwise. Conjugate transpose operation is A

Transpose without conjugation is A.

There are functional alternatives. x*y is the dot product of two column vectors.

cross(x,y) is the cross product when defined.


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Scalars and Matrices

A + x will add x to every entry in A

A*x will multiply every entry in A by x

A/x will divide in the same way.


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Data Analysis

max

min

mean median

std

var

sort

sum

prod cumsum

cumprod

diff


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Linear Algebra and MATLAB

norm(x,y)will give the y-norm of the

vector x. y = inf will produce the max

absolute value and y = -inf will produce themin absolute value.

The p-norm of a matrix is defined as

||Ax||p/||x||p, forxnon-zero in length.


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Linear Equations

The fundamental tool is the backslash

operator, \.

SolvesAx = b

AX = B


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Results

If A is n-by-n non-singular, A\b is the

solution to Ax = b. Solution methods

include:LU factorization with partial pivoting

Triangular (by substitution)

Hermitian positive definite (Choleskyfactorization)

Checks conditioning.


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Overdetermined

If A is m-by-n, with m>n, it has more

equations than unknowns.

A\b is a least squares solution.


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Underdetermined System

Fewer equations than unknowns.

If it is solvable at all, A\b produces a basic

solution. Otherwise A\b is a least squaressolution.


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Inverse, Pseudo-Inverse and

Determinant The matrix inverse is computed by inv(A).

Not usually needed, A\bis faster and more

accurate. det(A)is the determinant of a square

matrix. Sensitive to rounding errors, but

accurate for integer matrices. pinv(A)computes the pseudo-inverse of A.


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Tutorial Assignment

A perception of depth in a 2-D display can begenerated by the use of a stereogram.

Suppose you have some three-dimensional data{xi,yi,zi}.

Create a pair of side-by-side plots consisting of thevectors (x-d,y)and (x+d,y),where

di= c(zmax-zi)*(xmax-xmin)/(zmax-zmin). c depends on the separation of the displays and the

units of measurement. Explore various options.

Matrices Practice

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