MATH 410 Matrix Analysis
1.
Worksheet #6 07/15/2013
Which of the following formulas for v, w dene inner products on R2 ?
2
2
v1 + v2
2
2
w1 + w2
(a) v1 w2 + v2 w1
(b) (v1 + v2 )(w1 + w2 )
2.
(c)
(d) 2v1 w1 + (v1 v2 )(w1 w2 )
The unit circle for an inner p
(PM? @Amx §0LJWM
Math 410 (Prof. Bayly) EXAM 4: Monday 8 August 2005
There are 4 problems on this exam. They are not all the same length or difculty, nor the
same number of points. You should read through the entire exam before deciding which
problems you
2.1-2.4 Vector Spaces and related topics
10/20/14 8:52 AM
Vector Space: bunch of things (set of elements: set of vectors) which can be
added and multiplied by scalars
The whole vector space must be closed under these operations
o anything that can be obt
Operators and Matrices
10/20/14 8:52 AM
1) we say that a function (mapping), f, from
(
) is linear if
Linear maps are also called operators
Example:
Any m x n matrix A is an operator (linear map) from
because:
o
Example:
Matrices of Operators
1) Let
be a
L^(2)-optimization (least squares)
10/20/14 8:52 AM
Also called the best Linear Fit (or L 2 optimization)
Idea: introduce some error function and minimize it
Minimize the function, so take the derivative with respect to a0 and
a1
Suppose that matrix A is
MATH 410 Matrix Analysis
Worksheet #7 07/16/2013
1
2
1. Compute the 1-, 2-, 3-, and -norms for the vectors 2 and 1. In each case, verify the triangle inequality.
1
3
2. Which two of the vectors u = 2
2
1
T
,v= 1
4
1
T
,w= 0
0
1
T
are closest to each other
Matrix Analysis MATH 410 (Cais), Spring 2012
Homework 3: Due Monday, March 26 by 12:00pm
1. Use Cramers rule to solve the system of equations
2x + y + z = 3
xyz =0
x + 2y + z = 0
2. For each matrix below, nd the eigenvalues and corresponding eigenvectors.
Diagonalization
10/20/14 8:52 AM
Diagonalization:
Let A be an n x n matrix which has exactly n linearly independent
eigenvectors (or equivalently, all geometric multiplicities are equal
to algebraic multiplicities, or also equivalently, all geometric
mul
Jeremy Lerner
Math 410
23 April 2010
Extra credit: Euler Angles
Euler angles describe the angles at which the standard XYZ axes can be rotated.
The basis in the standard XYZ being the following e-vectors (basis vectors for the XYZ
axes):
"1 %
" 0%
"0%
! $
o Then:
o We know that if a matrix has no inverse if the determinant is
zero
o Characteristic Equation
Example
Eigenvectors:
Example:
Example for quiz:
Finding Eigenvectors
First: find the characteristic polynomial
o p(lambda)=det(A-(lambda)*I)
o Charact
Eigenvalues and Eigenvectors
10/20/14 8:51 AM
Functions of matrices:
This will teach us how to take the tangent, sine or cosine, or some
function of a matrix
Consider a square matrix (n x n) : A
Definition: we say that a number, lambda, is an eigenvalue
Change of Coordinates
10/20/14 8:52 AM
Example
Example
The action of A geometrically is stretching of the space in the
direction of v2 by a factor of 3
Example part 2
o
o Note: if an eigenvalue is negative, then you stretch the space
then reflect over the
Gram-Schmidt Orthogonalization Process
10/20/14 8:52 AM
A method to construct an orthonormal basis from an arbitrary basis
Example:
Using the given vectors construct an orthonormal basis
v2 is not orthogonal to v1 (we say that vectors x and y are
orthogo
Math 410: Matrix Analysis
10/20/14 8:51 AM
Quizzes on Fridays in class over one exact same homework problem (maybe
with different numbers) from last week or Monday of that week
Grading
Quizzes: 20% total of final grade
o 0: nothing
o 1: wrong answer/ ide
Functions of Matrices
10/20/14 8:52 AM
Applications to Differential Equations
Linear equations with constant coefficients for matrices
10/20/14 8:52 AM
o
To find a solution with a different initial condition
o
!
Remark: if we need to solve a vector equati
Determinants of Matrices
10/20/14 8:51 AM
Not following bookbe warned
Two dimensional example
lambda, A, B, C and D are some constant coefficients (A, B, C or D
could be zero)
d(u, v)=area of parallelogram spanned by u and v
Example:
o
o Note: We will use
@ (2,
"2,
'X,
ma WW 91w m; ' + ,
Math 410 (Prof. Bayly) EXAM 1: Monday 1 August 2005
There are 4 problems on this exam. They are not all the same length or difculty, nor the
same number of points. You should read through the entire exam before deciding
MATH 310 FALL 2015 (Prof. Bayly): SYLLABUS:
NOTE: The timing may vary from this proposed timetable; Make sure to stay up to date
with the course as it actually evolves.
M 24 Aug: Vectors, basic operations, length, angle, general and weighted norms
W 26 Au
Math 310 Fall 2015 (Prof. Bayly) Homework 1 Due Friday 28 August
Notational note: It can be a lot more convenient sometimes to write a vector as a row of numbers
instead of a column. The way we can remember that it really is a column is to use the transpo