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 Cn

[Def]

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

, then the order di 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 = x2

: 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 ui 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) = =

tr (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 Tii are square and = for all

[Cor] If , then x nonsingular s.t. x-1 A x =

where all are distinct

the integer

Note : ni is the algebraic multipicity of

If , then Range(xi) are invariant subspace

Jordan's Block : Ji =

= 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 :

J1 =

J2 =

the eigenspace of

the eigenspace of

Localized Eigenvalues and Derturbation

Gershgorin Circles : Di =

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. xi = 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 = ( dij )    C = ( cij ) 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) = x3 + 3 x2 + 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