Gemar 111012 Advisor Dr Rebaza Overview Definitions Theorems Proofs Examples Physical Applications Definition 1 We say that a subspace S or R n is invariant under A nxn or Ainvariant if ID: 674839
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Slide1
Schur Factorization
Heath
Gemar
11-10-12
Advisor: Dr.
RebazaSlide2
Overview
Definitions
Theorems
ProofsExamplesPhysical ApplicationsSlide3
Definition 1
We say that a subspace S or
R
n is invariant under Anxn, or A-invariant if:x ϵ S Ax ϵ
S,
Or equivalently, AS is a subset of S.Slide4
Definition 2
An m x n matrix Q is called Orthogonal if
Q
TQ=IProperties (m=n):
The columns of Q are orthonormal
The rows of Q are orthonormal
Example:
Slide5
Theorem 1
Let S be invariant under
A
nxn, with dim(S)=r. Then, there exists a nonsingular matrix Unxn such that:
Where T
11
is square of order r.
Proof:Let
be a basis of subspace S.
Therefore we can write:
For some scalars c
1
,…,cr.We can now expand β to form a basis of n vectors.Gram SchmidtHouseholder MatrixDefine with T11 of order r x r.
Slide6
Proof of Theorem 1 Continued:
Expansion of U leads to it being orthogonal.
Therefore
.
Define
Note: Au
1
=
λ
1
u1, where λ1 is a constant.
Slide7
Example
Let
, let
T
span S. We define
to be the expanded orthogonal basis.
Slide8
Theorem 2: Real
Schur
Factorization
Let Anxn be an arbitrary real matrix. Then, there exists an orthogonal matrix Qnxn, and a block upper triangular matrix T such that:
Where each diagonal block
T
ii
is either a 1x1 or 2x2 real matrix, the latter with a pair of complex conjugate eigenvalues. The diagonal blocks can be arranged in any prescribed order.
Slide9
Proof of Theorem 2
From proof of Theorem 1:
By Induction we know there exists a matrix W such that W
T
B
22
W is upper block triangular.
Define V=
diag(1,W) and Q=UV
Continue to redefine the lower right block until all new
eigenvalues
are determined.
Variation for complex
eigenvalues. Slide10
Example
Let
Eigenvalues
:
λ
1
=6.1429,
λ
2
=-7.9078,
λ
3,4
=-0.6175 ± 1.7365i.
Schur Factorization gives:
Slide11
Ordered Block Schur
Factorization
Let
Anxn be an arbitrary real matrix. Then, there exists an orthogonal matrix Qnxn, and a block upper triangular matrix T such that
Where T
11
is m x m and T
22
is (n – m) x (n – m), for some positive integer m. The diagonal blocks can be arranged in any prescribed order. Slide12
Example
Slide13
Continued…
Note: First two vectors of Q form an orthonormal basis of the vectors that span the negative
real eigenvalues.
Slide14
Physical Application
Huckel
Theory
Combination of MoleculesForm new basis Schur Factorization (Hamiltonian) New energy statesConnecting Orbits in Dynamical Systems
Remark:
Factorization is also possible for matrix A(t) where
are
as smooth as A(t).
Slide15
References
Rebaza
, Dr. Jorge. “A First Course in Applied Mathematics”.
Golub, Vanloan. “Matrix Computations”.Stewart, G.W. “Matrix Algebra”.Slide16
THANK YOU
Dr.
Rebaza
Dr. ReidMissouri State University Mathematics Department