Alg1231W8T2 - Eigenvectors and Eigenvalues of a reflection

Wednesday, 12 September 2018

9:35 PM

Machine generated alternative text:
-2 
6 
4 
2 
o 
-2 
2 
4 
x 
6 
Let's recall that a linear transformation T has an eigenvector v O precisely when T(v) = Av for some scalar which is called its eigenvalue_ 
We have seen that 
is the reflection in the line y 
y 
and •\'2 ¯ 
x _ This has eigenvalues Al — 1 
1 
O with respective eigenvectors 
and v 2 
Recall: the Maple notation for the vector 
is <1 , 
2 
We say that Al and are the eigenvalues of the matrix A 
Let's note that the sum of the eigenvalues is Al + — 
o 
1 
1 
O and the product of the eigenvalues is — 
O 
These two numbers, known as the trace 
and the determinant of A, play an interesting role in transformations of the plane.

 

Machine generated alternative text:
2 
-2 
R(v) 14 
4 
Let R be the linear map which reflects in R 
The other eigenvalue is — 
The matrix that represents T is 
The characteristic polynomial for B is 
Recall: 
1 
• the Maple notation for the matrix 
3 
1 
• the Maple notation for the vector 
2 
about the line y 
—x. Geometrically we can see that an eigenvector of R associated to the eigenvalue Al — 
1 is the vector 
O with associated eigenvector 
= det(B - ti) 
2 
4 
is

 

Machine generated alternative text:
2 
o 
-4 
--2 
-2 
S(v) 
2 
x 
4 
—4 
Consider the transformation S : IR2 R which is reflection in the origin Then S has matrix representation S(v) 
cv where 
The characteristic polynomial of C is 
One could say this has zeros Al — 
multiplicity 2 
p(t) = det(C — tI) = 
O but in such a case we prefer to say that C has a single eigenvalue 
O with 
In this case, an associated eigenvector for C is (pick all that apply) 
2 
1 
-2 
1 
1 
1 
2 
any non-zero vector in R 
2 
Cl any vector in IR 
1 
Recall: the Maple notation for the matrix 
3 
2 
112 >,< 314 
4

 

Machine generated alternative text:
Consider the vector space PI = {ax + b : a, b e R} , i.e. the space of polynomials of degree at most 1. Let T : PI PI be the map 
T(ax + b) :— 
Then 
T(4x + 7) — 
T(-3C+4) — 
4X-3 
If we identify these linear polynomials with vectors via 
and 
then T(ax + b) 
ax + b 
b 
and hence T has matrix representation 
o 
b 
1 
This matrix has characteristic polynomial 
p(t) = det(A — tI) 
From this we can determine that the linear transformation T has the set of eigenvalues 
eigenvectors {X+l ,x-l} 
Recall: 
• to enter a set like {1, 2} in Maple notation use the syntax {1, 2} 
• multiplication is denoted using an asterisk * 
1 
t"2-1 
{1,-1} 
O and, expressed in polynomial form, an associated set of 
• the vectors in the vector space PI are polynomials, so the vector 2x + 1 would be entered in the form 2*x+1.

 

Machine generated alternative text:
Consider the 4-dimensional vector space of 2 x 2 matrices 
a 
M2,2 (R) - 
b 
: a,b,c,deR 
d 
Define the transformation 
You may remember this matrix transformation is called the transpose: T(A) 
c 
AT Now 
b 
d 
It follows that T has 3 
eigenvalue 
—1. 
O linearly independent eigenvectors associated to the eigenvalue 
1 and 1 
O linearly independent eigenvectors associated to the 
1 
Recall: the Maple notation for the matrix 
3 
2 
112 >,< 314 
4

 

 

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