Minimal Polynomials and Jordan Normal Forms
1. Minimal Polynomials
Let A be an n n real matrix.
M1. There is a polynomial p such that p(A) = 0.
Proof. The space Mnn (R) of n n real matrices is an n2 -
MATH 245 Linear Algebra 2, Lecture Notes, by Stephen New
1. Ane Spaces, Convex Sets and Simplices
1.1 Note: In this section, we let F denote a xed eld (for example F = Q, R, C or Zp
with p prime) and
MATH 245 Linear Algebra 2, Solutions to Assignment 2
3
1: Let A = 2
1
1 2
4 4 M3 (R). Diagonalize A, if possible.
1 0
Solution: The characteristic polynomial of A is
3x
fA (x) = det(A xI) = det 2
1
1
MATH 245 Linear Algebra 2, Assignment 3
3 1 1
1 0 0
1: Let A =
0 1 1
1 3 1
Due Fri June 27
0
1
M4 (R). Put A into Jordan Form, if possible.
1
4
2: (a) Let A M10 (Q). Suppose the characteristic polyn
MATH 245 Linear Algebra 2, Solutions to the Midterm Test
[5]
1 4 2
1: Let A = 1 3 1 .
1 2 0
Find an invertible matrix P and a diagonal matrix D such that P 1A P = D.
Solution: The characteristic polyn
MATH 245 Linear Algebra 2, Assignment 2
3
1: Let A = 2
1
Due Fri June 6
1 2
4 4 M3 (R). Diagonalize A, if possible.
1 0
3 1 3
2: Let A = 2 4 6 M3 (R) and dene L : R3 R3 by L(x) = Ax. Let U be the subs
Math 245, Spring 2012
Assignment 5
Due Friday, June 15, in class.
1. Let J be a Jordan block with eigenvalue . Let A = etJ , where t C.
(a) Prove that
(A0 )ij =
tj i /(j i)!
0
if i j
otherwise .
(b) I
Math 245, Spring 2012
Assignment 10
Due Friday, July 20, in class.
1. Let x, y V , where V is a nite dimensional vector space over a eld F. Prove that x y =
y x in V V if and only if x and y are linea
Math 245, Spring 2012
Assignment 11
Not to be handed in.
1. Let cfw_v1 , v2 , . . . , vn be a basis for a vector space V over a eld F. Let cfw_f1 , f2 , . . . , fn be the
dual basis.
(a) Suppose 1 i
Math 245, Spring 2012
Assignment 9
Not to be handed in.
1. Let A Mnn (R) be a skew-symmetric matrix. Show that eA is an orthogonal matrix. (You
may use the fact that eX +Y = eX eY if XY = Y X .)
2. Le
Math 245, Spring 2012
Assignment 8
Due Friday July 6, in class.
1. Let A Mnn (F) be a self-adjoint matrix (where F = R or F = C). Prove that A is
positive-denite if and only if all eigenvalues of A ar
Family Name (Print):
Signature: :
ID. No.: #
Math 245 Midterm Examination
Tuesday Feb. 8, 2005
NO ADDITIONAL MATERIAL ALLOWED
Instructor C.T.Ng
Instructions:
1. Put your name, signature and ID.
MATH 235 Midterm Solutions Fall 2005
1. (a) [2] Given that det(A) = 2 and det(B) = 5, determine dct(A3B1ATB2).
52g: : dLHA'sBwgzx =1di+<mj5d1+mwdi+<mjl
OMB»
(b) [3] Let A be an n x n matrix. A i
Math 245 Homework 5 Solutions
1. Find two square matrices A and B such that (AB) 6= B A .
Solution: There are lots of answers to this, but the simplest one I know is:
1 0
1 0
A=
, B=
0 0
1 0
To check
The Fundamental Theorem of Linear Algebra
Author(s): Gilbert Strang
Source: The American Mathematical Monthly, Vol. 100, No. 9 (Nov., 1993), pp. 848-855
Published by: Mathematical Association of Ameri
Math 245 Homework 3 Solutions
1. Let T : V V be a unitary linear transformation on a finite dimensional
inner product space V . Prove that there is a unitary linear transformation
U : V V such that U
Math 245 Homework 2 Solutions
1. Let v R3 be a vector of length one. Prove that there exist real numbers
[0, 2] and [0, ] such that:
v = (sin cos , sin sin , cos )
[Hint: Think of as the angle betwee
Math 245 Homework 4
Due Wednesday, October 25
1. Consider the plane curve 41x2 24xy + 34y 2 = 25, whom we will call
Mary. Find real numbers A, B, and C such that there is a rigid motion of
the xy-plan
Math 245 Homework 4 Solutions
1. Consider the plane curve 41x2 24xy + 34y 2 = 25, whom we will call
Mary. Find real numbers A, B, and C such that there is a rigid motion of
the xy-plane transforming M
Math 245 Homework 6
Due 10:30am, Wednesday, November 15
1. Let V be a real vector space, and let T and U be linear transformations
from V to R. Define B : V V R by B(v, w) = T (v)U (w). Prove that
B i
Math 245 Homework 5
Due Wednesday, November 1
1. Find two square matrices A and B such that (AB) 6= B A .
2. Let T : V V be a linear transformation satisfying T T = 0. Prove
that T T = 0.
3. Let V and
http:/uwaterloo.oxdia.com
MATH 245 Midterm
Prof. K.R. Davidson
Tuesday, November 6, 2012 4:306:30
Marks
[8]
1. (a) Define the characteristic polynomial pT of T L(V ).
(b) State the Cayley Hamilton The
http:/uwaterloo.oxdia.com
MATH 245 Linear Algebra 2, Solutions to the Midterm Test, Spring 2013
[5]
1: Let A and B be non-empty convex sets in Rn . Let A + B = a + ba A, b B . Show
that A + B is con
University of
Math 245
WATERLOO
Final Examination
Linear Algebra II
Fall 2011
Instructor: Kenneth R. Davidson
Section: 001
Term: 1119
Materials: Ordinary scientic calculators
Date: Saturday, December
Math 245, Spring 2012
Assignment 7
Due Friday, June 29, in class.
Throughout this assignment, the eld F is either R or C.
1. A self-adjoint matrix A Mnn (F) is said to be positive-denite if the map :