Find the eigenvector (G-hI)v=0, when h=1. G=[1,0,1;0,1,1;1,1,2]*[v1,v2,v3]=[v1,v2,v3]

Let's solve the problem step by step.

### Problem Setup
We are given the equation:

\[
(G - hI) v = 0
\]

where:
- \( G \) is the given matrix.
- \( h = 1 \) (which is a scalar).
- \( I \) is the identity matrix.
- \( v \) is the eigenvector.

### Given Data
The matrix \( G \) is:

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

The scalar \( h = 1 \), and the identity matrix \( I \) for a 3x3 matrix is:

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

### Step 1: Compute \( G - hI \)
Subtract \( hI \) from \( G \):

\[
G - hI = \begin{pmatrix}
1 & 0 & 1 \\
0 & 1 & 1 \\
1 & 1 & 2
\end{pmatrix}
- \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
= \begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 1 \\
1 & 1 & 1
\end{pmatrix}
\]

### Step 2: Set Up the Equation \( (G - hI)v = 0 \)
Now we have:

\[
\begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 1 \\
1 & 1 & 1
\end{pmatrix}
\begin{pmatrix}
v_1 \\
v_2 \\
v_3
\end{pmatrix}
= \begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}
\]

### Step 3: Solve the System of Equations
This multiplication gives us the following system of linear equations:

\[
\begin{aligned}
0 \cdot v_1 + 0 \cdot v_2 + 1 \cdot v_3 &= 0 \quad \Rightarrow \quad v_3 = 0 \\
0 \cdot v_1 + 0 \cdot v_2 + 1 \cdot v_3 &= 0 \quad \Rightarrow \quad v_3 = 0 \\
1 \cdot v_1 + 1 \cdot v_2 + 1 \cdot v_3 &= 0 \quad \Rightarrow \quad v_1 + v_2 + v_3 = 0
\end{aligned}
\]

From the first two equations, \( v_3 = 0 \). Substitute \( v_3 = 0 \) into the third equation:

\[
v_1 + v_2 = 0 \quad \Rightarrow \quad v_1 = -v_2
\]

### Step 4: Eigenvector Solution
The eigenvector \( v \) can be expressed as:

\[
v = \begin{pmatrix}
v_1 \\
v_2 \\
v_3
\end{pmatrix}
= \begin{pmatrix}
-v_2 \\
v_2 \\
0
\end{pmatrix}
= v_2 \begin{pmatrix}
-1 \\
1 \\
0
\end{pmatrix}
\]

So, the eigenvector corresponding to \( h = 1 \) is a scalar multiple of:

\[
v = \begin{pmatrix}
-1 \\
1 \\
0
\end{pmatrix}
\]

### Summary
The eigenvector \( v \) for the given equation when \( h = 1 \) is any scalar multiple of:

\[
v = \begin{pmatrix}
-1 \\
1 \\
0
\end{pmatrix}
\]

Do you want further details or have any questions?

Here are some related questions:
1. How do you find eigenvectors for different values of \( h \)?
2. What is the geometric interpretation of an eigenvector?
3. How can eigenvectors be used in Principal Component Analysis (PCA)?
4. What is the relationship between eigenvalues and eigenvectors?
5. How do we normalize an eigenvector?

**Tip:** Always verify the solution by substituting the eigenvector back into the original equation to ensure it satisfies the condition \( (G - hI)v = 0 \).

Learn how to find the eigenvector v for the equation (G - hI)v = 0, where h = 1, using a step-by-step guide and matrix operations. Explore the matrix G, its subtraction with h times the identity matrix I, and solve the resulting system of equations to determine v. Suitable for advanced undergraduate students studying linear algebra and matrix theory.

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