construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible.
Just write down a diagonal matrix with one zero on the diagonal and then apply an orthogonal transformation. E.g. if you start with the matrix:
A =
[1 ,0
0,1]
And take the orthogonal transformation to be:
S =
[cos(theta) , -sin(theta)
sin(theta), cos(theta]
then
the transformed matrix is:
S A S^(-1)
S^(-1) =
[cos(theta) , sin(theta)
-sin(theta), cos(theta]
If you take theta = pi/4 you get the matrix:
1/2 [1,1
1,1]
To construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible, you can start with a diagonal matrix with one zero on the diagonal and then apply an orthogonal transformation.
Let's start with the matrix A:
A =
[1 , 0
0 , 0]
Notice that this matrix is diagonal but not invertible since it has a zero on its diagonal.
Next, we need to apply an orthogonal transformation. An orthogonal transformation can be represented by a 2 x 2 orthogonal matrix, denoted as S.
For simplicity, let's take the orthogonal transformation matrix S to be:
S =
[cos(theta) , -sin(theta)
sin(theta) , cos(theta)]
Now, we can compute the transformed matrix by multiplying A with S and its inverse:
SAS^(-1)
To find S^(-1), we can use the fact that an orthogonal matrix is also its own inverse, so S^(-1) = S^T.
S^(-1) =
[cos(theta) , sin(theta)
-sin(theta) , cos(theta)]
To obtain a diagonalizable but not invertible matrix, we can choose a specific value for theta. Let's take theta = pi/4.
After substituting the values, the transformed matrix becomes:
1/2 [1, 1
1, 1]
This matrix is diagonalizable because it can be written in the form PDP^(-1), where P is a matrix of eigenvectors and D is a diagonal matrix with eigenvalues on the diagonal. In this case, the eigenvalues are both equal to 1.
Therefore, the nondiagonal 2 x 2 matrix that is diagonalizable but not invertible is:
1/2 [1, 1
1, 1]
To construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible, follow these steps:
1. Start with the matrix A:
A = [1 , 0]
[0 , 1]
2. Define the orthogonal transformation matrix, S, as:
S = [cos(theta) , -sin(theta)]
[sin(theta) , cos(theta)]
3. Calculate the inverse of S, denoted as S^(-1):
S^(-1) = [cos(theta) , sin(theta)]
[-sin(theta) , cos(theta)]
4. Substitute the values of S and S^(-1) into the equation S * A * S^(-1):
S * A * S^(-1) = [cos(theta) , -sin(theta)] * [1 , 0] * [cos(theta) , sin(theta)]
[0 , 1]
= [cos(theta) , -sin(theta)] * [cos(theta) , sin(theta)]
[sin(theta) , cos(theta)]
5. Simplify the matrix multiplication:
Multiplying the first row of S with the first column of A gives:
[cos(theta) , -sin(theta)] * [cos(theta)] = cos^2(theta) - sin^2(theta)
Multiplying the first row of S with the second column of A gives:
[cos(theta) , -sin(theta)] * [sin(theta)] = cos(theta)sin(theta) - sin(theta)cos(theta) = 0
Multiplying the second row of S with the first column of A gives:
[sin(theta) , cos(theta)] * [1] = sin(theta)
Multiplying the second row of S with the second column of A gives:
[sin(theta) , cos(theta)] * [0] = 0
Therefore, the transformed matrix is:
[cos^2(theta) - sin^2(theta) , 0]
[0 , sin(theta)]
6. Choose a value for theta to obtain a nondiagonal, diagonalizable, and non-invertible matrix. For example, setting theta = pi/4:
With theta = pi/4, the transformed matrix becomes:
[1/2 , 0]
[0 , 1/2]
This is a nondiagonal 2 x 2 matrix that is diagonalizable (since it can be transformed into a diagonal matrix) but not invertible (since it has a zero eigenvalue, resulting in a determinant of zero).