Question

Let T₁ be the reflection about the line 4x?3y=04x?3y=0 in the euclidean plane. What is the standard matrix A of T₁ ?

What are the two eigenvalues and corresponding eigenspaces of A ?

Answer 1

We presume that the question mark means +.

The standard matrix for the transformation representing reflectiopn across the line y = mx is

(1-m2)/(1+m2)	2m/(1+m2)
2m/(1+m2)	-(1-m2)/(1+m2)

Here, the given line is 4x+3y = 0 or, y = -(4/3)x so that m = -4/3.

Then the standard matrix of T₁ is A =

-7/25	-24/25
-24/25	7/25

The characteristic equation of A is det(A-?I₂) = 0 or,?² -1 = 0 or,(?+1)(?-1) = 0. Thus, the 2 eigenvalues of A are ?₁ = 1 and ?₂= -1.

The eigenvector of A associated with its eigenvalue 1 is solution to the equation (A-I₂)X = 0. To solve this equation, we will reduce A-I₂ to its RREF which is

1	¾
0	0

Now, if X = (x,y)^T, then the equation (A-I₂)X = 0 is equivalent to x +3y/4 = 0 or, x = -3y/4 so that X = (-3y/4,y)^T = y(-3/4,1)^T. Thus, the eigenvector of A associated with its eigenvalue 1 is (-3/4,1)^T. The related eigenspace is span{(-3/4,1)^T }.

The eigenvector of A associated with its eigenvalue -1 is solution to the equation (A+I₂)X = 0. To solve this equation, we will reduce A+I₂ to its RREF which is

1	-4/3
0	0

Now, if X = (x,y)^T, then the equation (A+I₂)X = 0 is equivalent to x -4y/3 = 0 or, x = 4y/3. Then X = (4y/3,y)^T = y(4/3,1)^T. Thus, the eigenvector of A associated with its eigenvalue -1 is (4/3,1)^T. The related eigenspace is span{(4/3,1)^T }.