Question

Matrix ( A ) is factored in the form ( PDP^-1 ). Using the diagonalization theorem, how can the eigenvalues of ( A ) and a basis for each eigenspace be found?

A) By computing the determinant of ( A ) and its eigenvectors.
B) By performing row operations on ( A ) and finding its null space.
C) By diagonalizing ( A ) using ( P ) and ( D ) matrices.
D) By taking the transpose of ( A ) and its associated eigenvalues.

Answer 1

The correct answer is C) By diagonalizing (A) using (P) and (D) matrices. The diagonalization theorem states that a square matrix (A) can be diagonalized if it has (n) linearly independent eigenvectors, and the eigenvalues can be found by solving the characteristic equation. A basis for each eigenspace can be found by solving the system of equations (A - λI)x = 0.

Step-by-step explanation:

The correct answer is C) By diagonalizing (A) using (P) and (D) matrices.

The diagonalization theorem states that a square matrix (A) can be diagonalized if and only if it has (n) linearly independent eigenvectors, where (n) is the dimension of the matrix. If (A) is diagonalizable, it can be factored into the form (PDP^-1), where (P) is a matrix whose columns are the eigenvectors of (A), and (D) is a diagonal matrix with the eigenvalues of (A) on the diagonal.

To find the eigenvalues of (A), you can solve the characteristic equation det(A - λI) = 0, where (det) is the determinant, (λ) is the eigenvalue, and (I) is the identity matrix. The eigenvalues are the roots of the characteristic equation.

Once you have the eigenvalues, you can find a basis for each eigenspace by solving the system of equations (A - λI)x = 0, where (x) is a vector and (λ) is an eigenvalue. The solutions to this system will give you the eigenvectors of (A).