M = [[1, 2], [2, 1]], then then the eigenvector corresponding to its eigenvalue 3 is

Let's solve for the eigenvectors of the matrix \( M = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \) corresponding to its eigenvalue \( 3 \).

### Step 1: Eigenvalue-Eigenvector Equation
For a matrix \( M \) and an eigenvector \( v \) corresponding to an eigenvalue \( \lambda \), the equation is:

\[
M v = \lambda v
\]

Given \( M = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \) and \( \lambda = 3 \), we need to solve for \( v \).

This leads to the equation:

\[
\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = 3 \begin{bmatrix} x \\ y \end{bmatrix}
\]

This results in the following system of linear equations:

\[
\begin{aligned}
1x + 2y &= 3x \quad \text{(1)} \\
2x + 1y &= 3y \quad \text{(2)}
\end{aligned}
\]

### Step 2: Solve the System of Equations
1. From equation (1): \( 1x + 2y = 3x \)
   - Rearranging: \( 2y = 2x \)
   - Simplifying: \( y = x \)

2. From equation (2): \( 2x + 1y = 3y \)
   - Substituting \( y = x \) into this equation: \( 2x + x = 3x \), which is true and provides no additional information.

### Step 3: General Form of the Eigenvector
Since \( y = x \), the eigenvector corresponding to the eigenvalue \( 3 \) is of the form:

\[
v = \begin{bmatrix} x \\ x \end{bmatrix} = x \begin{bmatrix} 1 \\ 1 \end{bmatrix}
\]

Thus, the eigenvector corresponding to the eigenvalue \( 3 \) is any scalar multiple of \( \begin{bmatrix} 1 \\ 1 \end{bmatrix} \). Typically, it is written as:

\[
v = \begin{bmatrix} 1 \\ 1 \end{bmatrix}
\]

This is the eigenvector corresponding to the eigenvalue \( 3 \) for the matrix \( M \).

Would you like a detailed explanation of eigenvectors and their properties?

Here are 5 related questions for further exploration:
1. What is the eigenvector corresponding to the other eigenvalue of the matrix \( M \)?
2. How are eigenvalues and eigenvectors computed for larger matrices?
3. How do orthogonality properties relate to eigenvectors of symmetric matrices?
4. What are the applications of eigenvectors in quantum mechanics?
5. How can eigenvectors be used in stability analysis for dynamic systems?

**Tip:** Eigenvectors of symmetric matrices corresponding to different eigenvalues are orthogonal.

Learn how to find the eigenvector corresponding to eigenvalue 3 for the matrix M = [[1, 2], [2, 1]]. This problem involves solving the eigenvalue-eigenvector equation, understanding matrix operations, and applying linear algebra concepts. Suitable for advanced undergraduate students and professionals.

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