Give an example of a 3 x 3 matrix $A$ whose sole eigenvalue is 0, but which has only one linearly independent eigenvector $x$. What is the rank of $A$? A matrix all of whose sole eigenvalue is 0 is called nilpotent.
Added by Mohamed J.
Close
Step 1
This means the matrix is nilpotent and its minimal polynomial is $x^k$ for some $k \le 3$. Since the matrix is 3x3, the maximal possible value of $k$ is 3. A simple example of such a matrix is: $A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 Show more…
Show all steps
Your feedback will help us improve your experience
Victor Salazar and 86 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
An $n \times n$ matrix $A$ is called nilpotent if $A^{m}=0$ for some positive integer $m$. Examples are triangular matrices whose entries on the diagonal are all 0. Consider a nilpotent $n \times n$ matrix $A,$ and choose the smallest number $m$ such that $A^{m}=0 .$ Pick a vector $\vec{v}$ in $\mathbb{R}^{n}$ such that $A^{m-1} \vec{v} \neq \overrightarrow{0} .$ Show that the vectors $\vec{v}, A \vec{v}, A^{2} \vec{v}, \ldots, A^{m-1} \vec{v}$ are linearly independent Hint: Consider a relation $c_{0} \vec{v}+c_{1} A \vec{v}+c_{2} A^{2} \vec{v}+\cdots+$ $c_{m-1} A^{m-1} \vec{v}=\overrightarrow{0} .$ Multiply both sides of the equation with $A^{m-1}$ to show that $c_{0}=0 .$ Next, show that $c_{1}=0$ and so on.
Subspaces of $\mathbb{R}^{n}$ and Their Dimensions
The Dimension of a Subspace of $\mathbb{R}^{n}$
What does it mean for a matrix A to have an eigenvalue of 0, such as in Example 5? This happens if and only if the equation Ax = 0x has a nontrivial solution. But (4) is equivalent to Ax = 0, which has a nontrivial solution if and only if A is not invertible. Thus 0 is an eigenvalue of A if and only if A is not invertible. This fact will be added to the Invertible Matrix Theorem in Section 5.2.
Madhur L.
3. a) Find the eigenvalues of the following matrices. b) Find the eigenvector of corresponding eigenvalue, which is the minimum eigenvalue for the matrix A. Write down the corresponding eigenspace for this eigenvector. Hint: Do not find the other eigenvectors. Only one eigenvector in matrix A for the MIN eigenvalue.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD