LMM OPP MM M PP N - UAH - Engineering

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Chapter 3: Matrices - Elementary Theory

A matrix is a rectangular array of scalars.

L O a11 a12 L a1n

MMM PPP A =

a21 M

a22 M

L a2n M

NM QP am1 am2 L amn

John Stensby (3-1)

This matrix has m rows and n columns. Often, we use the notation {aij} to denote a matrix.

Representing a Linear Transformation by a Matrix

As given in Chapter 2, the definition of a linear transformation T : U V is somewhat

abstract. Once bases for spaces U and V are specified, a matrix can be used to provide a

definitive representation for a linear transformation.

Let U and V be vector spaces, over the same field F, of dimension n and m, respectively.

rr r

rr r

Let 1, 2, ... , n be a basis of U, and 1, 2, ... , m be a basis of V. Let T : U V be a linear

transformation. In terms of these bases, we can write

r m r

T( j) = aij i , 1 j n .

i=1

(3-2)

This equation defines an m?n matrix A that describes T : U V with respect to the given bases. As shown below, this matrix can be used as a representation for transformation T. Often, we use the symbolic equation

rr

r rr

r

T 1 2 L n = 1 2 L m A

(3-3)

rr

r

instead of the algebraic equation (3-2). The quantity 1 2 L n is itself an n?n matrix

rr r

with columns made from the basis vectors 1, 2, ... , n.

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John Stensby

r

r r

Given any X U, the matrix A can be used to compute Y = T(X) by using the coordinate

r r

r

vectors that represent X and Y with respect the bases. For X U, we can write

r X

=

n

x

j

r

j

,

j=1

(3-4)

r

where xi, 1 i n, are the coordinates for vector X. Now, use (3-4) to compute

F I r r G J Y = T(X) = T

n

x

j

r

j

=

n

x

j

r T(

j

)

H K j=1

j=1

(3-5)

Use (3-2) in (3-5) to obtain

F I F I r n m r m n

r mr

G J G J Y = xj aij i = aij xj i = yi i ,

H K H K j=1 i=1

i=1 j=1

i=1

(3-6)

where

n

yi = aij xj , 1 i m, j=1

(3-7)

r

are the coordinates of Y.

The algebraic derivation of (3-7) has a somewhat symbolic counterpart. First, recall that

r

X can be represented symbolically as

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L Ox1

M P r r MM PP X = 1

r 2

L

r n

x2 M

,

NMxn QP

Version 1.0

John Stensby (3-8)

r

where [x1 x2 ... xn]T is a coordinate vector for the vector X (remember: coordinates are in italics and vector components are not italicized). Note that the xk are just scalars, and apply T to (3-8) to obtain

L Ox1

M P r

r

d i MM PP T X = T 1

r 2

L

r n

x2 M

(3-9)

NMxn QP

Now, use (3-3) in (3-9) to obtain

L O L O x1

y1

M P M P r r d i MM PP MM PP T X = [1

r 2

L

r m ]A

x2 M

r = [1

r 2

L

r m ]

y2 M

,

NMxn QP

NM yn QP

(3-10)

where

L MMM

y1 y2 M

O PPP

=

A

L MMM

x1 x2 M

O PPP

=

L MMM

a11 a21 M

a12 a22

M

L L

a1n a2n

OPPLMMxx12

O PP

M PM M P

(3-11)

NMynQP NMxnQP NMam1 am2 L amnQPNMxnQP

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r r

is the desired relationship between the coordinate vectors used to represent X and Y. Equations

(3-7) and (3-11) state equivalent results: the rows of m?n matrix A are used to express the

r

r

coordinate vector for Y in terms of the coordinate vector for X.

Example

r

r

Let U = V = R2. Let 1 = [1 0]T and 2 = [0 1]T be a

r T(X)

r X

common basis for U, V. Define T as the transformation /2

that rotates vectors counterclockwise by /2 radians.

Note that T does not change the magnitude of the vector.

Observe that

F THG

L1OI NM0QPKJ

=

L0O NM1QP

=

0

L1O NM0QP

+

1

L0O NM1QP

and

F THG

L0OI NM1QPKJ

=

L-1O NM 0 QP

=

-1

L1O NM0QP

+

0

L0O NM1 QP

so that

A

=

L0 NM1

-1O 0 QP

is a matrix representation for the rotation transformation. As shown above, there is a well defined relationship between a linear transformation and

the matrix used to represent the transformation. Hence, almost all of the definitions, given in Chapter 2, dealing with linear transformations have counterparts when dealing with matrices. For example, let T : U V be represented by matrix A. Then the vectors in R(T) are represented by coordinate vectors in the subspace

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LMMMLMMM OPPP LMMM OPPP OPPP R(A) =

y1

x1

y2 M

=A

x2 M

where [x1 x2 L xn ]Tis a coordinate vector in U

NMMNMymQP NMxmQP

QPP

John Stensby (3-12)

Clearly, R(A) is the span of the columns of A. Likewise, the vectors in K(T) are represented by coordinate vectors in the subspace

L M MM K(A) = NMM

LM M

x1 x2

OP P

MMP

NMxn QP

:

A

LM M

x1 x2

OP P

MMP

NMxn QP

L0O

=

MM0PP MMP

NM0QP

OP QPPPP

(3-13)

Matrices: Some Special Kinds In our work (and most applications), the elements of a matrix belong to the field of real

numbers, denoted as R, or the field of complex numbers, denoted as C. If A is an m?n matrix over the real numbers we say A Rm?n. Likewise, if A is an m?n matrix over the complex numbers we say A Cm?n.

An n?n matrix P Rn?n is said to be orthogonal if PT = P-1 so that PTP = P-1P = I, the n?n identity matrix. The columns (and rows) of Q are orthogonal. Let

rr

r

Q = q1 q2 L q n ,

(3-14)

then Q is orthogonal if and only if

rr qi,q j

=

r q

T j

r qi

=1

i= j

=0

i j

(3-15)

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