Problem 6. Check wether it is possible to orthogonally diagonalise the matriz 3
ID: 3136667 • Letter: P
Question
Problem 6. Check wether it is possible to orthogonally diagonalise the matriz 3 -21 A= 1-2 6-2 1 -2 3 Write dowm this diagonalisation erplicitly if it is possible. Problem 7. Let A be the matriz from the previous problem. Write doun an expression for the quadratic form f(x, y, z) = xTAx, trhere x = y in terms of x, y, or indefinite? . Is this quadratic form positive definite, negative definite Write dourn a basis B, and numbers 1, X2,A3 such that f can be represented as where X, Y, Z are coordinates in the basis B. Write down the transition matrix from coordinates in this basis B to standard coordinates. Evaluate f(x, y, 2) at one unit eigenvector corresponding to each eigenvalue.Explanation / Answer
6. We have A =
3
-2
1
-2
6
-2
1
-2
3
The characteristic equation of A is det(A-I3) = 0 or, 3-122+36-32 = 0 or, (-8)(-2)2 = 0. Thus, the eigenvalues of A are 1=8 and 2,3=2. The eigenvectors of A associated with its eigenvalue 2 are solutions to the equation (A-2I3)X = 0. To solve this equation, we will reduce the matrix A-2I3 to its RREF which is
1
-2
1
0
0
0
0
0
0
Thus, if X = (x,y,z)T, then the equation (A-2I3)X = 0 is equivalent to x-2y+z0 or, x = 2y-z so that X = (2y-z,y,z)T = y(2,1,0)T+z(-1,0,1)T. Thus, v2 = (2,1,0)T and v3 = (-1,0,1)T are the eigenvectors of A associated with its eigenvalue 2. Similarly v1= (1,-2,1)T is the eigenvector of A associated with its eigenvalue 8.
Now, let D =
8
0
0
0
2
0
0
0
2
and P =
1
2
-1
-2
1
0
1
0
1
Then A = PDP-1
Please post problem 7 again separately.
3
-2
1
-2
6
-2
1
-2
3