vector norms def: a norm is a function that satisfies p-norms: the most important class of vector...

Vector Norms DEF: A norm is a function R R n : that satisfies p-norms: The most important class of vector norms Example: 3 5 2 x 1 x 10 2 x 1644 . 6 9 25 4 x 5 p x p p p p 3 5 2

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Vector Norms

DEF: A norm is a function RRn : that satisfies

p-norms: The most important class of vector norms

Example:

1x 10

2x 1644.69254

x 5

x p ppp 352

Page 2: Vector Norms DEF: A norm is a function that satisfies p-norms: The most important class of vector norms Example:

Example: }1:{ xRxS n

}1:{2 xRx n

}1:{1 xRx n

}1:{ xRx n

}1:{ p

n xRx

Vector Norms

Matrix Norm Induced by Vector Norm

DEF: the matrix norm of A (induced by the vector norm) is defined to be

Rxnm x

AxA

sup mx

nmAxA

,sup

DEF: If the matrix A is a square matrix

nAxA

sup

Page 4: Vector Norms DEF: A norm is a function that satisfies p-norms: The most important class of vector norms Example:

Matrix Norm Induced by Vector Norm

DEF: If the matrix A is a square matrix

nAxA

sup

Example:

21A

The unit vector x that is amplified most by A is [0,1]^T, the amplification factor is 4.

41A

Page 5: Vector Norms DEF: A norm is a function that satisfies p-norms: The most important class of vector norms Example:

Matrix Norm Induced by Vector Norm

DEF: If the matrix A is a square matrix

nAxA

sup

Example:

21A

The unit vector x that is amplified most by A is the vector indicated by the dashed line, the amplification factor is 2.9208. 9208.2

Page 6: Vector Norms DEF: A norm is a function that satisfies p-norms: The most important class of vector norms Example:

Holder Inequalities

Rem

nnnxAAx

Cauchy-Schwarz:

22xxyxT

Holder Inequality:

T xxyx

qp,1 111

Page 7: Vector Norms DEF: A norm is a function that satisfies p-norms: The most important class of vector norms Example:

Holder Inequalities

Example:Tn uvARu,v and Let

2 :Compute A

Page 8: Vector Norms DEF: A norm is a function that satisfies p-norms: The most important class of vector norms Example:

Bounding Norm of Product

BOUND:

,R and ,R,Ron norms be and, , mnl

mnl

matrix an be andmatrix be nmBmlA

nmmlnlBAAB

,,,

Example: matrix nnA

mmAA

Page 9: Vector Norms DEF: A norm is a function that satisfies p-norms: The most important class of vector norms Example:

Frobenius norm or Hilbert-Schmidt

DEF: Let A be a mxn matrix2

1 1

jijFaA

REM:

jiFaA

A ofcolumn th -j ja

)( AAtrA T

Page 10: Vector Norms DEF: A norm is a function that satisfies p-norms: The most important class of vector norms Example:

Frobenius norm or Hilbert-Schmidt

BOUND:nmmlnl

BAAB,,,

BOUND:FFF

BAAB

Proof:

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