91Ó°ÊÓ

Eigenvectors are special vectors associated with a matrix. When a matrix acts on an eigenvector, it simply scales that vector by a certain amount, called the eigenvalue. So, for a matrix \( A \) and an eigenvector \( \mathbf{v} \) with eigenvalue \( \lambda \), the relationship is given by:
\( A\mathbf{v} = \lambda \mathbf{v} \).
Eigenvectors provide important insights into the structure of a matrix. They help in various applications like stability analysis, quantum mechanics, and even search engine algorithms.
In our case:
- The eigenvalues \( r_1 = 1 \) and \( r_2 = -1 \) correspond to the eigenvectors \( \mathbf{v}_1 = \begin{bmatrix} 1 & 1 \end{bmatrix}\) and \( \mathbf{v}_2 = \begin{bmatrix} 0 & 1 \end{bmatrix} \) respectively.

A linear combination of vectors involves multiplying each vector by a scalar and then adding the results. This is a way to express one vector as a combination of other vectors. In the context of our problem:
- We express the vector \( \mathbf{w} = \begin{bmatrix} 2 & 3 \end{bmatrix} \) as a linear combination of the eigenvectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \).
- To do so, we solve
\[ \mathbf{w} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 \], finding \( c_1 \) and \( c_2 \).
- This expresses \( \mathbf{w} \) as \( \mathbf{w} = 2\mathbf{v}_1 + \mathbf{v}_2 \) after finding \( c_1 = 2 \) and \( c_2 = 1 \).
This step is crucial because it allows us to utilize the eigenvectors' properties to simplify matrix operations.

The eigenvalue property is fundamental in simplifying matrix operations like raising a matrix to a power. For any eigenvector \( \mathbf{v}_i \) of a matrix \( A \) with eigenvalue \( r_i \), the following holds:
- \( A\mathbf{v}_i = r_i\mathbf{v}_i \)
When we raise the matrix to the power \( n \), the property extends to:
- \( A^n \mathbf{v}_i = r_i^n \mathbf{v}_i \).
In our problem:
- Since \( r_1 = 1 \) and \( r_2 = -1 \), then
\( A^{100} \mathbf{v}_1 = 1^{100} \mathbf{v}_1 = \mathbf{v}_1 \) and
\( A^{100} \mathbf{v}_2 = (-1)^{100} \mathbf{v}_2 = \mathbf{v}_2 \).
This property significantly simplifies the computation of \( A^n \mathbf{w} \).

Matrix exponentiation is the process of raising a matrix to a given power. It often involves multiplying the matrix by itself repeatedly. However, using eigenvalues and eigenvectors, we can greatly simplify this process.
- Given a matrix \( A \) and its eigenvectors and eigenvalues, \( A^n \) can be expressed efficiently.
For our problem:
1. Decompose \( \mathbf{w} \) into eigenvectors
2. Use the eigenvalue property:

\( A^{100} \mathbf{w} = A^{100} (2\mathbf{v}_1 + \mathbf{v}_2) = 2A^{100} \mathbf{v}_1 + A^{100} \mathbf{v}_2 \)
3. Simplify using the eigenvalue property to get:
\( A^{100} \mathbf{w} = 2\mathbf{v}_1 + \mathbf{v}_2 = \begin{bmatrix} 2 & 3 \end{bmatrix} \).
This allows for an efficient computation of matrix powers and their effects on vectors.