### Instruction: What is the principal component for a dataset with the following covariance matrix?
```
Covariance matrix = [[1, 0.9], 
                    [0.9, 1]]
```
Find the eigenvalue and eigenvector of the covariance matrix and calculate the principal component.### Response: To find the principal component, we first need to find the eigenvalues and eigenvectors of the covariance matrix. The covariance matrix is given as:

Covariance matrix = [[1, 0.9], 
                    [0.9, 1]]

Let's find the eigenvalues (λ) by solving the characteristic equation, which is the determinant of (Covariance matrix - λI) = 0, where I is the identity matrix.

|1-λ, 0.9  | = 0
|0.9, 1-λ|

Expanding the determinant, we get:

(1-λ)(1-λ) - (0.9)(0.9) = 0
λ^2 - 2λ + 1 - 0.81 = 0
λ^2 - 2λ + 0.19 = 0

Now, we can use the quadratic formula to find the eigenvalues:

λ = (2 ± sqrt((-2)^2 - 4(1)(0.19))) / 2(1)
λ = (2 ± sqrt(4 - 0.76)) / 2
λ = (2 ± sqrt(3.24)) / 2

The two eigenvalues are:

λ1 = (2 + sqrt(3.24)) / 2 ≈ 1.62
λ2 = (2 - sqrt(3.24)) / 2 ≈ 0.38

Now, let's find the eigenvectors for each eigenvalue by solving the following equation:

(Covariance matrix - λI) * eigenvector = 0

For λ1 = 1.62:

|1-1.62, 0.9  | * |x1| = |0|
|0.9, 1-1.62|   * |x2|   |0|

|-0.62, 0.9 | * |x1| = |0|
|0.9, -0.62|   * |x2|   |0|

Solving this system of linear equations, we get the eigenvector:

eigenvector1 = [1, 1]

For λ2 = 0.38:

|1-0.38, 0.9  | * |x1| = |0|
|0.9, 1-0.38|   * |x2|   |0|

|0.62, 0.9 | * |x1| = |0|
|0.9, 0.62|   * |x2|   |0|

Solving this system of linear equations, we get the eigenvector:

eigenvector2 = [1, -1]

The principal component is the eigenvector corresponding to the largest eigenvalue. In this case, the largest eigenvalue is λ1 = 1.62, and its corresponding eigenvector is [1, 1]. Therefore, the principal component is [1, 1].