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Just explain to me how he turned the eigen value 1 into a vector 0,1,0 

To explain how the eigenvalue \(\lambda_1 = 1\) led to the eigenvector \(\mathbf{v}_1 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\), let’s break it down step by step.

### Step 1: Eigenvalue Equation
Given the matrix \(A\) and the eigenvalue \(\lambda_1 = 1\), we use the standard eigenvalue equation:
\[
A\mathbf{v}_1 = \lambda_1 \mathbf{v}_1
\]
This becomes:
\[
A\mathbf{v}_1 = 1 \cdot \mathbf{v}_1
\]
or equivalently:
\[
(A - I)\mathbf{v}_1 = 0
\]
where \(I\) is the identity matrix.

### Step 2: Substituting \(A\) and \(\lambda_1\)
The matrix \(A\) is given as:
\[
A = \begin{pmatrix} 4 & 0 & 1 \\ -2 & 2 & 0 \\ -2 & 0 & 1 \end{pmatrix}
\]
Subtracting \(I\) (the identity matrix) multiplied by the eigenvalue \(\lambda_1 = 1\):
\[
A - I = \begin{pmatrix} 4-1 & 0 & 1 \\ -2 & 2-1 & 0 \\ -2 & 0 & 1-1 \end{pmatrix}
= \begin{pmatrix} 3 & 0 & 1 \\ -2 & 1 & 0 \\ -2 & 0 & 0 \end{pmatrix}
\]

### Step 3: Solve \((A - I)\mathbf{v}_1 = 0\)
Now, we solve the homogeneous system:
\[
\begin{pmatrix} 3 & 0 & 1 \\ -2 & 1 & 0 \\ -2 & 0 & 0 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}
\]
This system gives the following equations:
1. \(3x_1 + x_3 = 0\)  (from the first row)
2. \(-2x_1 + x_2 = 0\)  (from the second row)
3. \(-2x_1 = 0\)  (from the third row)

### Step 4: Solve for \(x_1, x_2, x_3\)
From the third equation, \(-2x_1 = 0\), we see that \(x_1 = 0\).

Substitute \(x_1 = 0\) into the second equation:
\[
-2(0) + x_2 = 0 \quad \Rightarrow \quad x_2 = 0
\]

Now substitute \(x_1 = 0\) into the first equation:
\[
3(0) + x_3 = 0 \quad \Rightarrow \quad x_3 = 0
\]

So the eigenvector associated with \(\lambda_1 = 1\) is \(\mathbf{v}_1 = \begin{pmatrix} 0 \\ 1 \\ 0 \end}.


Just explain to me how he turned the eigen value 1 into a vector 0,1,0

This problem involves finding the eigenvector associated with the eigenvalue λ=1 for a given 3x3 matrix. Learn the step-by-step solution to solve the equation (A - I) * v = 0 and find the eigenvector v = (0, 1, 0).

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