Matrix Theory, Math6304
Lecture Notes from Sept 11, 2012
taken by Tristan Whalen
1
Further Review continued
Warm-up
Let A, B Mn and suppose det(A) = 0. Dene a matrix valued function as follows:
F (t) = (A + tB )1
The determinant of F (t) is just a polynom
Matrix Theory, Math6304
Lecture Notes from October 02, 2012
taken by Sanat Kumar Upadhyay
Last time 09/27/2012
Review
Real matrices
Warm up:
Let O Mn (R) be an orthogonal matrix (i.e. a rotation matrix). Then by denition OT O =
OOT = In , so O is unitary.
Matrix Theory, Math6304
Lecture Notes from September 27, 2012
taken by Tasadduk Chowdhury
Last Time (09/25/12):
QR factorization: any matrix A Mn has a QR factorization: A = QR, where Q is unitary
and R is upper triangular. In addition, we proved that if
Matrix Theory, Math6304
Lecture Notes from October 4, 2012
taken by Yuricel Mondragn
o
Last Time (10/2/12):
Amendment
We should clean up the proof of the basis property from last time. To see that Bj 1 is a basis,
note that
dim ker A|vj1 = dim spancfw_An1
Matrix Theory, Math6304
Lecture Notes from October 9, 2012
taken by Charles Mills
Last Time (10/6/12)
Suppose A Mn satises A = A , then given A, we can dene a map qA : Cn R by
qA (x) =< Ax, x > . It is clear that if we know the matrix A, we know qA and vi
Matrix Theory, Math6304
Lecture Notes from October 23, 2012
taken by Satish Pandey
Warm up from last time
Example of low rank perturbation; re-examined
We had this operator
1
0
. .
.
.
S=
Mn ( C)
.
. 1
1
0
S is said to be the Cyclic shift operator in the
Matrix Theory, Math6304
Lecture Notes from October 16, 2012
taken by Ricky Ng
4.3
Estimation of eigenvalues for sums of Hermitian matrices (continued)
Last time we saw some useful applications of the Courant-Fischer Theorem. Namely, we obtain
a lower and
Matrix Theory, Math6304
Lecture Notes from October 11, 2012
taken by Da Zheng
4
Variational characterization of eigenvalues, continued
We recall from last class that given a Hermitian matrix, we can obtain its largest (resp. smallest)
eigenvalue by maximi
Matrix Theory, Math6304
Lecture Notes from September 25, 2012
taken by Katie Watkins
Last Time (9/20/12)
Cayley Hamilton
Block diagonalization with triangular blocks
Nearly Diagonalizability
Q: We saw Schur does not give a unique upper triangular form.
Matrix Theory, Math6304
Lecture Notes from September 20, 2012
taken by Ilija Jegdic
Last Class
Unitary diagonalization and normality
Warm up
If A is normal, and A = B + iC , B = B , C = C what we can conclude about B , C ?
We have that
AA = (B + iC ) (B +
Matrix Theory, Math6304
Lecture Notes from August 30, 2012
taken by Andy Chang
Last Time (8/28/12)
Course info: website - math.uh.edu/bgb
Matrix multiplication: left (premultiplication) and right (postmultiplication)
Nullity and rank: dimension counting
D
Matrix Theory, Math6304
Lecture Notes from August 28, 2012
taken by Bernhard Bodmann
0
Course Information
Text: R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1885.
Oce: PGH 604, 713-743-3851, Mo 2-3pm, We 1-2pm
Email: bgb@math.uh.ed
Matrix Theory, Math6304
Lecture Notes from September 4, 2012
taken by Thomas Weber
Last Time (8/30/12)
Gram-Schmidt: - set of linearly independent vectors can yield an orthonormal set that spans
the space
Trace and Determinant: - denitions and properties
Matrix Theory, Math6304
Lecture Notes from September 6, 2012
taken by Nathaniel Hammen
Last Time (9/4/12)
Diagonalization: conditions for diagonalization
Eigenvalue Multiplicity: algebraic and geometric multiplicity
1
1.1
Further Review
Warm-up questions
Matrix Theory, Math6304
Lecture Notes from September 18, 2012
taken by John Haas
Last Time (9/13/12)
Unitary Matrices: denition and some characterizations/results
Householder Transforms : basic computation
Unitary Equivalence: denition and some conditions
Matrix Theory, Math6304
Lecture Notes from September 13, 2012
taken by Manisha Bhardwaj
Last Time (9/11/12)
Invariant Subspaces: denition
Commuting families: eigenspaces and simultaneous diagonalization
Hermitian matrices: denition and characterization
1
Matrix Theory, Math6304
Lecture Notes from October 30, 2012
taken by Thomas Weber
Last Time (10/25/12)
Eigenvalue interlacing: general eigenvalue interlacing for principal submatrices
Rayleigh-Ritz principle: generalized Rayleigh-Ritz principle for sum of