Finding eigenvalues and eigenvectors

The eigenvalues of a square matrix are defined by the condition that there be a nonzero solution to the homogeneous equation $(A-\lambda I)\vvec=\zerovec$.

1. If there is a nonzero solution to the homogeneous equation $(A-\lambda I)\vvec = \zerovec$, what can we conclude about the invertibility of the matrix $A-\lambda I$?

2. If there is a nonzero solution to the homogeneous equation $(A-\lambda I)\vvec = \zerovec$, what can we conclude about the determinant $\det(A-\lambda I)$?

3. Let's consider the matrix from which we co…

In this activity, we will find the eigenvectors of a matrix as the null space of the matrix $A-\lambda I$.

1. Let's begin with the matrix $A = \left[\begin{array}{rr} 1 & 2 \\ 2 & 1 \\ \end{array}\right]$. We have seen that $\lambda = 3$ is an eigenvalue. Form the matrix $A-3I$ and find a basis for the eigenspace $E_3 = \nul(A-3I)$. What is the dimension of this eigenspace? For each of the basis vectors $\vvec$, verify that $A\vvec = 3\vvec$.

2. We also saw that $\lambda = -1$ is an eigenvalue. Form the matrix $A-(-1)I$ and find a basis for the eigenspace

1. Identify the eigenvalues, and their multiplicities, of an $n\times n$ matrix whose characteristic polynomial is $(2-\lambda)^3(-3-\lambda)^{10}(5-\lambda)$. What can you conclude about the dimensions of the eigenspaces? What is the shape of the matrix? Do you have enough information to guarantee that there is a basis of $\real^n$ consisting of eigenvectors?

2. Find the eigenvalues of $\left[\begin{array}{rr} 0 & -1 \\ 4 & -4 \\ \end{array}\right]$ and state their multiplicities. Can you find a basis of $\real^2$ consisting of eig…

For each of the following matrices, find its characteristic polynomial, its eigenvalues, and the multiplicity of each eigenvalue.

1. $A=\left[\begin{array}{rr} 4 & -1 \\ 4 & 0 \\ \end{array}\right]$.

2. $A=\left[\begin{array}{rrr} 3 & -1 & 3 \\ 0 & 4 & 0 \\ 0 & 0 & -6 \end{array}\right]$.

3. $A = \left[\begin{array}{rr} -2 & 0 \\ 0 & -2 \\ \end{array}\right]$.

4. $A=\left[\begin{array}{rr} -1 & 2 \\ 2 & 2 \\ \end{array}\right]$.

Determine whether the following statements are true or false and provide a justification for your response.

1. The eigenvalues of a matrix $A$ are the entries on the diagonal of $A$.

2. If $\lambda$ is an eigenvalue of multiplicity $1$, then $E_\lambda$ is one-dimensional.

3. If a matrix $A$ is invertible, then $\lambda=0$ cannot be an eigenvalue.

4. If $A$ is a $13\times 13$ matrix, the characteristic polynomial has degree less than $13$.

5. The eigenspace $E_\lambda$ of $A$ is t…

Provide a justification for your response to the following questions.

1. Suppose that $A$ is a $3\times 3$ matrix having eigenvalues $\lambda = -3,3,-5$. What are the eigenvalues of $2A$?

2. Suppose that $D$ is a diagonal $3\times 3$ matrix. Why can you guarantee that there is a basis of $\real^3$ consisting of eigenvectors of $D$?

3. If $A$ is a $3\times 3$ matrix whose eigenvalues are $\lambda = -1,3,5$, can you guarantee that there is a basis of $\real^3$ consisting of eigenvectors of $A$?

4. Suppose t…

For each of the following matrices, use Sage to determine its eigenvalues, their multiplicities, and a basis for each eigenspace. For which matrices is it possible to construct a basis for $\real^3$ consisting of eigenvectors?

1. $A = \left[\begin{array}{rrr} -4 & 12 & -6 \\ 4 & -5 & 4 \\ 11 & -20 & 13 \\ \end{array}\right]$

2. $A = \left[\begin{array}{rrr} 1 & -3 & 1 \\ -4 & 8 & -5 \\ -8 & 17 & -10 \\ \end{array}\right]$

3. $A = \left[\begin{array}{rrr} 3 & -8 & 4 \\ -2 & 3 & -2 \\ -6 & 12 & -7 \\ \end{array}\right]$

There is a relationship between the determinant of a matrix and the product of its eigenvalues.

1. We have seen that the eigenvalues of the matrix $A = \left[\begin{array}{rr} 1 & 2 \\ 2 & 1 \\ \end{array}\right]$ are $\lambda = 3,-1$. What is $\det A$? What is the product of the eigenvalues of $A$?

2. Consider the triangular matrix $A = \left[\begin{array}{rrr} 2 & 0 & 0 \\ -1 & -3 & 0 \\ 3 & 1 & -2 \\ \end{array}\right]$. What are the eigenvalues of $A$? What is $\det A$? What is the product of the eigenvalues of $A$?

3. Based on these examples, what do you think is the relationship between th…

Consider the matrix $A=\left[\begin{array}{rr} 0.5 & 0.6 \\ -0.3 & 1.4 \\ \end{array}\right]$.

1. Find the eigenvalues of $A$ and a basis for their associated eigenspaces.

2. Suppose that $\xvec_0=\twovec{11}{6}$. Express $\xvec_0$ as a linear combination of eigenvectors of $A$.

3. Define the vectors . Write $\xvec_1$, $\xvec_2$, and $\xvec_3$ as a linear combination of eigenvectors of $A$.

4. What happens to $\xvec_k$ as $k$ grows larger and larger?

Consider the matrix $A=\left[\begin{array}{rr} 0.4 & 0.3 \\ 0.6 & 0.7 \\ \end{array}\right]$

1. Find the eigenvalues of $A$ and a basis for their associated eigenspaces.

2. Suppose that $\xvec_0=\twovec{0}{1}$. Express $\xvec_0$ as a linear combination of eigenvectors of $A$.

3. Define the vectors . Write $\xvec_1$, $\xvec_2$, and $\xvec_3$ as a linear combination of eigenvectors of $A$.

4. What happens to $\xvec_k$ as $k$ grows larger and larger?