Activity 4.2.4

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?

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 eigenvectors of this matrix?

Consider the matrix $A = \left[\begin{array}{rrr} -1 & 0 & 2 \\ -2 & -2 & -4 \\ 0 & 0 & -2 \\ \end{array}\right]$ whose characteristic equation is $$(-2-\lambda)^2(-1-\lambda) = 0$$.

1. Identify the eigenvalues and their multiplicities.

2. For each eigenvalue $\lambda$, find a basis of the eigenspace $E_\lambda$ and state its dimension.

3. Is there a basis of $\real^3$ consisting of eigenvectors of $A$?

Now consider the matrix $A = \left[\begin{array}{rrr} -5 & -2 & -6 \\ -2 & -2 & -4 \\ 2 & 1 & 2 \\ \end{array}\right]$ whose characteristic equation is also $$(-2-\lambda)^2(-1-\lambda) = 0$$.

1. Identify the eigenvalues and their multiplicities.

2. For each eigenvalue $\lambda$, find a basis of the eigenspace $E_\lambda$ and state its dimension.

3. Is there a basis of $\real^3$ consisting of eigenvectors of $A$?

Consider the matrix $A = \left[\begin{array}{rrr} -5 & -2 & -6 \\ 4 & 1 & 8 \\ 2 & 1 & 2 \\ \end{array}\right]$ whose characteristic equation is $$(-2-\lambda)(1-\lambda)(-1-\lambda) = 0$$.

1. Identify the eigenvalues and their multiplicities.

2. For each eigenvalue $\lambda$, find a basis of the eigenspace $E_\lambda$ and state its dimension.

3. Is there a basis of $\real^3$ consisting of eigenvectors of $A$?