# 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.

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:

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).