Question

Consider the length 3 recurrence \(x_{k+3}=a x_{k}+b x_{k+1}+c x_{k+2}\) a. If \(\mathbf{v}_{k}=\left[\begin{array}{c}x_{k} \\ x_{k+1} \\\ x_{k+2}\end{array}\right]\) and \(A=\left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c\end{array}\right]\) show that \(\mathbf{v}_{k+1}=\bar{A} \mathbf{v}_{k}\) b. If \(\lambda\) is any eigenvalue of \(A,\) show that \(\mathbf{x}=\left[\begin{array}{c}1 \\ \lambda \\\ \lambda^{2}\end{array}\right]\) is a \(\lambda\) -eigenvector. [Hint: Show directly that \(A \mathbf{x}=\lambda \mathbf{x}\).] c. Generalize (a) and (b) to a recurrence $$ x_{k+4}=a x_{k}+b x_{k+1}+c x_{k+2}+d x_{k+3} $$ of length 4 .

Short answer

Extend the logic to the 4-length recurrence with vector \(\mathbf{w}_k\) and matrix \(B\), confirming eigenvector properties similarly.

Step-by-step solution

Define the matrix-vector product for part (a)

To show that \(\mathbf{v}_{k+1} = A \mathbf{v}_{k}\), we plug in \(\mathbf{v}_{k}\) and calculate this matrix product: \[ \mathbf{v}_{k+1} = \left[\begin{array}{c} x_{k+1} \ x_{k+2} \ x_{k+3} \end{array}\right] = A \left[\begin{array}{c} x_{k} \ x_{k+1} \ x_{k+2} \end{array}\right]. \] Substituting in matrix \(A\), we have \[ A = \left[\begin{array}{ccc}0 & 1 & 0 \ 0 & 0 & 1 \ a & b & c\end{array}\right] \] and therefore, \(A \mathbf{v}_{k} \) becomes \[ \left[\begin{array}{ccc}0 & 1 & 0 \ 0 & 0 & 1 \ a & b & c\end{array}\right] \left[\begin{array}{c} x_{k} \ x_{k+1} \ x_{k+2} \end{array}\right] = \left[\begin{array}{c} x_{k+1} \ x_{k+2} \ a x_{k} + b x_{k+1} + c x_{k+2} \end{array}\right]. \] This is exactly \(\mathbf{v}_{k+1}\), confirming the relation.

Determine eigenvector condition for part (b)

We need to show that if \(\lambda\) is an eigenvalue of \(A\), then \(\mathbf{x} = \left[\begin{array}{c} 1 \ \lambda \ \lambda^{2} \end{array}\right]\) is an eigenvector corresponding to \(\lambda\). Start by applying \(A\) to \(\mathbf{x}\): \[ A\left[\begin{array}{c} 1 \ \lambda \ \lambda^{2} \end{array}\right] = \left[\begin{array}{ccc} 0 & 1 & 0 \ 0 & 0 & 1 \ a & b & c \end{array}\right] \left[\begin{array}{c} 1 \ \lambda \ \lambda^{2} \end{array}\right]. \] Calculating the product, the result is \[ \left[\begin{array}{c} \lambda \ \lambda^{2} \ a(1) + b(\lambda) + c(\lambda^{2}) \end{array}\right]. \] Under the condition that \(a + b\lambda + c\lambda^{2} = \lambda^{3}\), this simplifies to \[ \left[\begin{array}{c} \lambda \ \lambda^{2} \ \lambda^{3} \end{array}\right] = \lambda \left[\begin{array}{c} 1 \ \lambda \ \lambda^{2} \end{array}\right], \] confirming \(\mathbf{x}\) is a \(\lambda\)-eigenvector.

Generalize for recurrence of length 4 for part (c)

Extend the previous concepts to handle a recurrence of the form \(x_{k+4} = ax_{k} + bx_{k+1} + cx_{k+2} + dx_{k+3}\). Define a new vector \(\mathbf{w}_{k} = \left[\begin{array}{c} x_{k} \ x_{k+1} \ x_{k+2} \ x_{k+3} \end{array}\right]\) and matrix \(B\): \[ B = \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \ a & b & c & d \end{array}\right]. \] You can show \(\mathbf{w}_{k+1} = B \mathbf{w}_{k}\) using similar logic as in Step 1, with \[ B \left[\begin{array}{c} x_{k} \ x_{k+1} \ x_{k+2} \ x_{k+3} \end{array}\right] = \left[\begin{array}{c} x_{k+1} \ x_{k+2} \ x_{k+3} \ ax_{k} + bx_{k+1} + cx_{k+2} + dx_{k+3} \end{array}\right]. \] For part (b), if \(\mu\) is an eigenvalue of \(B\), an analogous eigenvector is \(\left[\begin{array}{c} 1 \ \mu \ \mu^{2} \ \mu^{3} \end{array}\right]\).

Key concepts

Eigenvectors and Eigenvalues

Understanding eigenvectors and eigenvalues is crucial in linear algebra and computer graphics. They help us understand how transformations behave, especially when dealing with recurrence relations and matrix operations.
When we say that a vector \( \mathbf{x} \) is an eigenvector of a matrix \( A \), it means that applying \( A \) to \( \mathbf{x} \) results in a scalar multiple of \( \mathbf{x} \). This scalar is called an eigenvalue, typically denoted by \( \lambda \).
The mathematical expression for this relationship is:

• \( A\mathbf{x} = \lambda \mathbf{x} \)

In essence, eigenvectors indicate directions that do not change during a transformation, only their scale changes by factor \( \lambda \).
In computer graphics, eigenvectors can be used to determine invariant lines, which remain unaltered despite various transformations, crucial for creating stable and predictable graphics.
Identifying these vectors and their corresponding eigenvalues often involves solving characteristic equations, which can sometimes be complex, but vital for understanding how systems evolve over time.

Matrix-Vector Multiplication

Matrix-vector multiplication is fundamental in linear algebra, playing a pivotal role in computer graphics and numerical simulations. It involves transforming a vector space through matrix operations.
Given a vector \( \mathbf{v} \) and a matrix \( A \), the product \( A \mathbf{v} \) is another vector. The calculation involves:

• Each element of the resulting vector involves a dot product of the respective row of the matrix with the vector.
• This process is repeated for each row, generating a complete transformed vector.

Consider matrix \( A \) of size \( m \times n \) and vector \( \mathbf{v} \) of size \( n \). The resulting vector will have \( m \) components derived from the interactions described above. This operation not only changes the position but also scales the vector \( \mathbf{v} \), useful for graphics transformations like scaling, rotation, and translation.
In the context of the provided exercise, matrix-vector multiplication is used to express complex recurrence relations as linear transformations, making it easier to comprehend and solve the evolution of vector sequences.

Recurrence Relations

Recurrence relations are sequences defined using previous terms. They are found recurrently in algorithms, numerical methods, and especially in computer graphics for iterative shape or pattern generation.
A simple linear recurrence can be expressed in the form:

• \( x_{k+1} = a x_k + b x_{k-1} + \, \ldots\, \)

This means that each term is a linear combination of prior terms, governed by a specific rule.
In higher dimensions, they can be understood through matrix equations, where transformation matrices dictate the iterative process. This converts the problem into a more manageable linear algebra form.
The benefit of expressing them in matrix form is the ability to use powerful tools from linear algebra like eigenvectors and eigenvalues to analyze stability and behavior of such sequences, crucial in applications like predicting recursive graphics patterns or solving complex differential equations iteratively.