hw problem

Step 1: Find the eigenvalues

To find the eigenvalues, we need to solve the characteristic equation, which is given by

det (A - 1J) = 0, where A is the given matrix, 1

represents the eigenvalues, and I is the identity matrix.

For the given matrix, we have:

1-A0 111-11101-시=

Expanding the determinant along the second column:

(1 - 1) 1-111 1-1=0

(1 - 1)((1 - 1)2- 1) = 0

（1 入）（1-2入+入2-1）=0

(1 - 1) (12 - 21) = 0

(1 - 1) 1(1- 2) =0

Thus, the eigenvalues are 11 = 0, 12 = 1, and 13 = 2.

Step 2: Find the eigenspace

Step 2: Find the eigenspace for lambda 1= 0_

To find the eigenspace, we solve the system

(A - 11I)v = 0, where v is the eigenvector.

1-00 111-0 1101-00

1

0 1 1

111 0 1xY z=100

From the equations, we have:

x+z=0

x +y+z=0

So,

Z =—X

and x + y - x = 0, thus y = 0. The eigenvector is

of the form |x 0 - x. A basis for the eigenspace is 1 0 - 1.

Step 3: Find the eigenspace for

Step 5: Diagonalization

Since we have three distinct eigenvalues for a 3x3 matrix, the matrix is diagonalizable. The matrix P is formed by using the eigenvectors as columns:

P=1 0 1012-10

1|

The diagonal matrix D is formed by placing the eigenvalues on the diagonal:

D= 10 0001000

2]

Then, A = PDP-1.

Step 6: Final Answer

The eigenvalues are 0, 1, and 2. The bases for the eigenspaces are 1 0 - 11, l0 1 0l, and (1 2 1]. The matrix A is diagonalizable.

1. What is the characteristic equation used to find the eigenvalues of a matrix?

2. How do you represent the eigenvalues in the characteristic equation?

3. What steps are involved in expanding the determinant to find eigenvalues?

4. Given the matrix's eigenvalues, what does each eigenvalue represent in terms of the matrix's properties?

5. How do you find the eigenspace associated with a given eigenvalue?

6. In the process of finding the eigenspace, what equation must be solved?

7. What forms the basis for the eigenspace when finding it for the eigenvalue λ = 0?

8. What steps lead to determining the relationships between the variables x, y, and z in the eigenspace?

9. Why is it important to find the eigenvalues and eigenspaces of a matrix?

10. Can you explain the significance of the eigenvector found in the eigenspace calculation?

11. What indicates that a matrix is diagonalizable?

12. How is the matrix P constructed in the diagonalization process?

13. What is the significance of the eigenvectors used as columns in matrix P?

14. How is the diagonal matrix D formed, and what does it contain?

15. What does the equation A = PDP^{-1} represent in the context of diagonalization?

16. Why is diagonalization important for understanding the properties of a matrix?

1. The characteristic equation used to find the eigenvalues of a matrix is given by det(A - λI) = 0, where A is the matrix, λ represents the eigenvalues, and I is the identity matrix.

2. The eigenvalues in the characteristic equation are represented by the symbol λ.

3. The steps involved in expanding the determinant to find eigenvalues include subtracting λI from the matrix A, and calculating the determinant of the resulting matrix, setting it equal to zero.

4. Each eigenvalue represents a scalar indicating how much a corresponding eigenvector is stretched or compressed during the transformation represented by the matrix. It also provides insights into the properties of the matrix, such as stability and the nature of solutions in differential equations.

5. The eigenspace associated with a given eigenvalue is found by solving the equation (A - λI)v = 0, where v is the eigenvector.

6. The equation that must be solved in the process of finding the eigenspace is (A - λI)v = 0.

7. The basis for the eigenspace for the eigenvalue λ = 0 can be represented as the vector of the form (v = \begin{pmatrix}x \\ 0 \\ -x \end{pmatrix}) for any scalar x. A basis can be chosen, for example, as (\begin{pmatrix}1 \\ 0 \\ -1 \end{pmatrix}).

8. The steps leading to determining the relationships between the variables x, y, and z in the eigenspace are derived from the system of equations obtained from solving (A - λI)v = 0, which provides expressions connecting these variables.

9. Finding the eigenvalues and eigenspaces of a matrix is important because they provide critical insight into the behavior of linear transformations, help in solving systems of differential equations, and are essential in various applications, including stability analysis and system dynamics.

10. The significance of the eigenvector found in the eigenspace calculation is that it represents a direction in which the transformation associated with the matrix acts as a simple scaling operation (i.e., the eigenvector does not change direction after transformation).

11. A matrix is indicated to be diagonalizable if it has a complete set of linearly independent eigenvectors corresponding to its eigenvalues.

12. The matrix P in the diagonalization process is constructed by using the eigenvectors as its columns.

13. The eigenvectors used as columns in matrix P form the basis for the transformation that diagonalizes the matrix, allowing for simplification of matrix operations.

14. The diagonal matrix D is formed by placing the eigenvalues on the diagonal, with zeros elsewhere. It contains the eigenvalues that correspond to the eigenvectors in P.

15. The equation A = PDP^{-1} represents the relationship where A is the original matrix, P is the matrix formed from its eigenvectors, D is the diagonal matrix with eigenvalues, and P^{-1} is the inverse of P, indicating how A can be expressed in a simplified diagonal form.

16. Diagonalization is important for understanding the properties of a matrix because it simplifies many matrix operations, reveals fundamental characteristics of the transformation, and can greatly ease calculations involving powers of a matrix as