Chapter 4.4 Similarity Transformation and Relative Topics

[`Def`] A
B ( similar ) if
P nonsingular s.t.
*A* =
*P*^{-1} *B P* ( where
)

[`Thm`] Similar matrices have the same Eigenvalues

<**Proof**> If
nonsingular s.t.
*A* = *P*^{-1} *B P*
Consider (
) =
=
=
=
=
=

[`Thm`] If *v* is an Eigenvector of A with associated Eigenvalues

If
*B* = *P*^{-1} *A P* , then
(*P*^{-1} *V*) is an Eigenvector of B with associated

<**Proof**>
, and
*B* = *P*^{-1} *A P*
*P B P*^{-1} *v* =
=

[`Def`]
is simple if the
Eigenvectors of A form a basis for *C*^{n}

[`Def`]

1. Algebraic multiplicity :
if the characteristic polynomial of A is given by
*f*(*x*) =

, then the order *d*_{i} is called the algebraic multiplicity of
2.
=
, then the geometric multiplicity =

Note : Geometric multiplicity Algebraic multiplicity

*Example : *
Rank =
Nullity =

Characteristic polynomial =
( *x* - 1 )^{2}

( ) : Algebraic multiplicity = 2 : geometric multiplicity = 2

Characteristic polynomial = *x*^{2}

: Algebraic multiplicity = 2

: geometric multiplicity = 1

[`Thm`] Similar Transformation preserve both the algebraic
and geometric multiplicities

Companion matrix

then = ( Jodan Form , Rational Form )

[`Thm`] ( diagonalizable )

If
is simple with *l.I.*
eigenvectors

then there exists
where
*u*_{i} are column vectors s.t.
*D* = *u*^{-1} *A u*
, where *D* =

( i.e. )

where f : characteristic poly. of A P : minimal poly. of A

Unitary Similary Transformation

A similar to B ( )

if nonsingular

s.t.
*A* = *P*^{-1} *B P*

( )

[`Def`]
is unitary if *U U*^{*} = *I* ( i.e.
*U*^{*} = *U*^{-1} )

where *U*^{*} is the conjugate transpose of *U*

Note 1.: If *U* unitary and
is orthogonal

unitary matrix , then A is unitary similary to B

( in is called orthogonal similary )

[`Def`] A is Hermite , if

( in is called symmetric )

Note 2.: Unitary matrix have the same properties as the orthogonal matrix
*Example : *
(i.)
(*vx* , *vy* ) = ( *x* , *y* ) = *y*^{*} *x* (ii.)
(iii.) If
are unitary

also unitary
(iv.) *A* = *VR* ( the same to
)

the set of all eigenvalues

If =

*det* (*A*) =
=

*t*_{r} (*A*) =

S is invariant ( of A ) , if

Note : x is eigenvector of A

( eigenspace of x is invariant ( of A ))

Note : If invariant

If

the same to

*Ax* = *xB* is called the Similary Transformation

( Note : If nonsingular )

[*Lemma*] If
satisfy
, then
unitary s.t.

Note : 1. 2. This Q is the Q of analysis x = QR

*Example : *

<**Proof**>
is the QR factorization of x

Because

( because R is nonsingular )

[`Schur's Theorem`]
If
, then there exist Q unitray
s.t.
*Q*^{*} *A Q* = *T* = *D* + *N* , where *D* =
are eigenvalues of A

Moreover can be rearranged in any order

<**Proof**> (By Induction on n )

n = 1 ( get ) ( )

then from the Lemma , unitary

s.t. If the Thm is hold for all order n - 1

unitary s.t.

Let

then *Q*^{*} *A Q* =
=
*T* = *D* + *N* =

*Example : *
then

*Q*^{*} *A Q* = *T* =
=

Note : If A is nonsingular N = 0

[`Def`]
departure from nonsingularity

N is independent choice of Q

Note : Simple matrices are dense in ( or ) ( i.e. If A is choosen randomly slmost all A are simple )

If are simple matrices

for any s.t.

[`Def`]
is said to be normal if
*A A*^{*} = *A*^{*} *A*

[`Thm`] ( Spectral Thm for Normal Matrices )

is normal
unitary , s.t.
*Q*^{*} *A Q* = *D* , D diagonal

<**Proof**> "
"
*Q*^{*} *A Q* = *D*

*D D*^{*} = (*Q*^{*} *A Q*)(*Q*^{*} *A Q*)^{*} = *Q*^{*} *A A*^{*} *Q*

=
*Q D D*^{*} *Q*^{*} = *A A*^{*}

and

" " A is normal T is normal

s called departure from the normality

A is normal if
*A A*^{*} = *A*^{*} *A*

[`Thm`] A is normal
unitary s.t.
*v*^{*} *A v* = *D*

Note : 1. ( Schur's Decomposition ) 2.

*Example : *

Note : Simple matrices are dense in .

For all is simple

Note : Matrices norm are equivalent , i.e. are two matrices norm s.t.

Note :
is indep. of choice of Q .
*Q A Q* = *D* + *N*

*Example : *
( Jordan Form of A )

Nonunitary Transformation

If
, *A* =
*p* + *q* = *n*

[`Def`]
s.t.
, then
is nonsingular

If
is nonsingular and *Y* is defined by

, then
*Y*^{-1} *A Y* =

[`Thm`] If *A* =
,
*Y s*.*t*. (*YQ*)^{-1} *A* (*YQ*)=

where *T*_{ii} are square and
=
for all

[*Cor*] If
, then
x nonsingular s.t.
*x*^{-1} *A x* =

where all are distinct

the integer

Note : *n*_{i} is the algebraic multipicity of

If
, then
*Range*(*x*_{i}) are invariant subspace

Jordan's Block : *J*_{i} =

= Note : Transformation is called "nonunitary transformation" .

*Example : *
,
*f*(*x*) =
( *t* - 2 )^{4} ( *t* - 3 )^{3} ,
*P*(*x*) =
( *t* - 2 )^{2} ( *t* - 3 )^{3}

Jordan Form :

*J*_{1} =

*J*_{2} =

the eigenspace of

the eigenspace of

Localized Eigenvalues and Derturbation

Gershgorin Circles :
*D*_{i} =

*Example : *

Gershgorin Circles are :

[`Thm`] ( Gershgorin Thm )

If , then the eigenvalues of A contained the main union of the Gershgorin circles

*Example : *

Note :

Let x be an Eigenvector ( )

s.t.
Let *i* be the index , s.t. *x*_{i} = 1

[`Thm`] If A is diagonized by a similary transform
*P*^{-1} *A P*

and *E* be any matrix , then the eigenvalues of

*A*+*E* lies in the union of

where

*Example : *
If
*D* = *P*^{-1} *A P* ,
and *A* is perturbed to *E* , then if

and , ,

<**Proof**> If
*D* = *P*^{-1} *A P*
,then
=
=
=
Let
*C* = *P*^{-1} *E P* , and
*D* = ( *d*_{ij} ) *C* = ( *c*_{ij} )
then ( From the Gershgorin's Thm )

=

=

= =

If A is Hermitian ( i.e.
*A* = *A*^{*} ) ( or Symmetric if
)

then A is unitary similary to a diagonal matrix D

i.e.
unitary ( orthogonal ) s.t.
*D* = *v*^{*} *A v*

Note : ( or are column vectors )

1. 2.

**Exercise 4.2.2 : **
If
and *P*(*X*) : char. poly. ,
then *P*(*X*) = 0

*Example : *

*P*(*X*) = *x*^{3} + 3 *x*^{2} + 3 *x* + 1

= = = 0

[`Thm`] Real Schur's Thm ( Schur's Thm
*v*^{*} *A v* = *T* = *D* + *N* )

s.t.
*Q*^{*} *A Q* = *T*
=

where or