Find the generalized  eigen vector of rank 3 ,corresponding the eigen valur 7 for the matrix 7 1 2  0 7 1      0 0 7

We are tasked with finding the generalized eigenvector of rank 3 corresponding to the eigenvalue \( \lambda = 7 \) for the matrix:

\[
A = \begin{pmatrix} 
7 & 1 & 2 \\ 
0 & 7 & 1 \\ 
0 & 0 & 7 
\end{pmatrix}
\]

### Step 1: Eigenvalue Equation
The eigenvalue is given as \( \lambda = 7 \). We know that for an eigenvalue \( \lambda \), the generalized eigenvectors \( v \) satisfy:

\[
(A - \lambda I)^k v = 0
\]

Where \( I \) is the identity matrix, and \( k \) is the smallest integer such that the equation holds. For a rank 3 generalized eigenvector, we need \( (A - 7I)^3 v = 0 \) but \( (A - 7I)^2 v \neq 0 \) and \( (A - 7I)v \neq 0 \).

### Step 2: Compute \( A - 7I \)

\[
A - 7I = \begin{pmatrix} 
7 & 1 & 2 \\ 
0 & 7 & 1 \\ 
0 & 0 & 7 
\end{pmatrix} - \begin{pmatrix} 
7 & 0 & 0 \\ 
0 & 7 & 0 \\ 
0 & 0 & 7 
\end{pmatrix}
= \begin{pmatrix} 
0 & 1 & 2 \\ 
0 & 0 & 1 \\ 
0 & 0 & 0 
\end{pmatrix}
\]

### Step 3: Find the Generalized Eigenvector

Let \( N = A - 7I \), then:

\[
N = \begin{pmatrix} 
0 & 1 & 2 \\ 
0 & 0 & 1 \\ 
0 & 0 & 0 
\end{pmatrix}
\]

We are looking for a vector \( v \) such that:

\[
N^3 v = 0, \quad N^2 v \neq 0, \quad Nv \neq 0
\]

#### Step 3.1: Generalized Eigenvector of Rank 1 (Eigenvector)

First, find the eigenvector corresponding to the eigenvalue 7, i.e., \( v_1 \) such that:

\[
Nv_1 = 0
\]

The null space of \( N \) is spanned by vectors of the form \( v_1 = (1, 0, 0)^T \), because \( N(1, 0, 0)^T = (0, 0, 0)^T \).

#### Step 3.2: Generalized Eigenvector of Rank 2

Next, we find \( v_2 \) such that:

\[
N v_2 = v_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}
\]

Solve the system \( Nv_2 = v_1 \), i.e.,

\[
\begin{pmatrix} 
0 & 1 & 2 \\ 
0 & 0 & 1 \\ 
0 & 0 & 0 
\end{pmatrix} 
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}
=
\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}
\]

This gives the following system of equations:

\[
x_2 + 2x_3 = 1 \\
x_3 = 0
\]

So, \( x_3 = 0 \) and \( x_2 = 1 \). Therefore, \( v_2 = (x_1, 1, 0)^T \). For \( Nv_2 = v_1 \), we can take \( v_2 = (0, 1, 0)^T \).

#### Step 3.3: Generalized Eigenvector of Rank 3

Finally, we find \( v_3 \) such that:

\[
N v_3 = v_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}
\]

Solve \( Nv_3 = v_2 \):

\[
\begin{pmatrix} 
0 & 1 & 2 \\ 
0 & 0 & 1 \\ 
0 & 0 & 0 
\end{pmatrix} 
\begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}
=
\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}
\]

This gives the system:

\[
y_2 + 2y_3 = 0 \\
y_3 = 1
\]

So, \( y_3 = 1 \) and \( y_2 = -2 \). Thus, \( v_3 = (y_1, -2, 1)^T \). For \( Nv_3 = v_2 \), we can take \( v_3 = (1, -2, 1)^T \).

### Conclusion:

The generalized eigenvector of rank 3 corresponding to the eigenvalue \( 7 \) is:

\[
v_3 = \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}
\]

Would you like more details or explanations on any part? Here are some follow-up questions:

1. How do you interpret the role of rank in generalized eigenvectors?
2. What are the geometric implications of generalized eigenvectors?
3. Can we apply a similar approach to other matrices?
4. How would you verify that this is indeed the correct generalized eigenvector?
5. What role do Jordan forms play in generalized eigenvector analysis?

**Tip:** Always check if you can simplify the system of equations when finding generalized eigenvectors to reduce computation complexity.

Find the generalized eigenvector of rank 3, corresponding to the eigenvalue 7 for the matrix 7 1 2  0 7 1  0 0 7

Learn how to find the generalized eigenvector of rank 3 for the eigenvalue 7 in a 3x3 matrix. The solution involves calculating the eigenvalue equation, computing the null space, and identifying the generalized eigenvector step by step. Suitable for college-level linear algebra courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation