Lecture 2 Orthogonal Vectors and Matrices, Norms

Lecture 2

Orthogonal Vectors and Matrices, Norms

MIT 18.335J / 6.337J

Introduction to Numerical Methods

Per-Olof Persson

September 12, 2006

1

Transpose and Adjoint

? For real A, the transpose of A is obtained by interchanging rows/columns

a11 a12

A

=

a21

a22

=

AT

=

a11

a21

a31

a12 a22 a32

a31 a32

? The adjoint or hermitian conjugate also takes complex conjugate

a11 a12

A

=

a21

a22

=

A

=

a11

a12

a21 a22

a31

a32

a31 a32

?

If real A = AT , then A is symmetric. If A = A, then A is hermitian.

2

Inner Product

? Inner product of two column vectors x, y Cm

m

xy = xiyi

i=1

? Euclidean length of x

x = xx =

m

1/2

|xi|2

i=1

? Angle between x, y

cos = xy xy

3

In MATLAB

Quantity

MATLAB Syntax Comment

Transpose AT

A.'

Transpose only

Adjoint A

A'

Transpose + complex conjugate

Inner product xy x'*y dot(x,y)

or '* assumes column vectors

Length x

sqrt(x'*x) or '* assumes column vector norm(x)

4

Orthogonal Vectors

? The vectors x, y Rm are orthogonal if xy = 0

? The sets of vectors X, Y are orthogonal if every x X is orthogonal to every y Y

? A set of (nonzero) vectors S is orthogonal if vectors pairwise orthogonal, i.e., for x, y S, x = y xy = 0

and orthonormal if, in addition,

every x S has x = 1

5

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