Multivariate Analysis
Content
3.
Let A be a diagonalizable matrix and satisfy A4 = A. (1) Find the possible eigenvalues of A.
Answer: A is a diagonalizable matrix so A can be defined as like:
where P can be form from eigenvector of matrix A and D is a diagonal matrix with eigenvalue of matrix A as the main diagonal. It is said that:
Because D is a diagonal matrix with eigenvalue of matrix A as the main diagonal, it can be written too like:
=
[
]
[
]
The only possible nonzero eigenvalues (λ) that can satisfy that equation is:
So the only possible nonzero eigenvalues of A is 1. (2) Answer: Prove and disprove that A2 = A.
Because D is a diagonal matrix with eigenvalue of matrix A as the main diagonal, it can be written too like:
[
]
[
]
Therefore,
[
]
[
]
The equation is satisfy therefore it can be concluded (3) Determine the rank of A.
Answer: For diagonalizable matrix A, the rank will be equal to the number of nonzero eigenvalues. The number of nonzero eigenvalues of matrix A is 1, which 1. , so the rank of A will be
Let A be a diagonalizable matrix and satisfy A4 = A. (1) Find the possible eigenvalues of A.
Answer: A is a diagonalizable matrix so A can be defined as like:
where P can be form from eigenvector of matrix A and D is a diagonal matrix with eigenvalue of matrix A as the main diagonal. It is said that:
Because D is a diagonal matrix with eigenvalue of matrix A as the main diagonal, it can be written too like:
=
[
]
[
]
The only possible nonzero eigenvalues (λ) that can satisfy that equation is:
So the only possible nonzero eigenvalues of A is 1. (2) Answer: Prove and disprove that A2 = A.
Because D is a diagonal matrix with eigenvalue of matrix A as the main diagonal, it can be written too like:
[
]
[
]
Therefore,
[
]
[
]
The equation is satisfy therefore it can be concluded (3) Determine the rank of A.
Answer: For diagonalizable matrix A, the rank will be equal to the number of nonzero eigenvalues. The number of nonzero eigenvalues of matrix A is 1, which 1. , so the rank of A will be
