Problem 3 - A Computational Introduction to Number Theory and Algebra

Question

EXERCISE \(18.3 .\) This exercise develops an alternative characterization of linearly generated sequences. Let \(\Psi=\left\\{z_{i}\right\\}_{i=0}^{\infty}\) be a sequence of elements of \(F\). Further, suppose that \(\Psi\) has minimal polynomial \(\phi=\sum_{j=0}^{m} c_{j} X^{j}\) with \(m>0\) and \(c_{m}=1\). Define the matrix $$ A:=\left(\begin{array}{cccc} z_{0} & z_{1} & \cdots & z_{m-1} \\ z_{1} & z_{2} & \cdots & z_{m} \\ \vdots & \vdots & \ddots & \vdots \\ z_{m-1} & z_{m} & \cdots & z_{2 m-2} \end{array}\right) \in F^{m \times m} $$ and the vector $$ w:=\left(z_{m}, \ldots, z_{2 m-1}\right) \in F^{1 \times m} $$ Show that $$ v=\left(-c_{0}, \ldots,-c_{m-1}\right) \in F^{1 \times m} $$ is the unique solution to the equation $$ v A=w . $$.

Step-by-Step Solution

Answer

Question: Prove that the vector \(v=\left(-c_{0}, \ldots,-c_{m-1}\right) \in F^{1 \times m}\) is the unique solution to the equation \(vA=w\), where \(A\) is an \(m\times m\) matrix whose elements are \(z_i\), and \(w\) is an \(1 \times m\) vector with elements \(z_{m}, \ldots, z_{2 m-1}\) and the minimal polynomial \(\phi=\sum_{j=0}^{m} c_{j} X^{j}\), where \(m>0\) and \(c_m=1\). Answer: We can prove this by following two steps: 1. Show that \(vA=w\) by substitution. By analyzing the given sequence and minimal polynomial, we found that \((vA)_j = -c_0z_j - c_1z_{j-1} - \cdots - c_{m-1}z_{j-m+1}\), which satisfies the relation \(vA=w\) for \(j=m, \ldots, 2m-1\). 2. Prove the uniqueness of the solution. By assuming another vector \(v'\) such that \(v'A=w\) and analyzing the product \((v'-v)A=0\), we found that \(v'\) should also obtain its coefficients from the minimal polynomial \(\phi\). However, this contradicts the assumption since we already deduced that \(v\) is the unique solution with coefficients negated from \(\phi\). Therefore, \(v\) is indeed the unique solution to the equation \(vA=w\).

Step 1: Show \(vA=w\) by substitution

Since \(z_i\) are elements of a linearly generated sequence, they satisfy the minimal polynomial, i.e., \(\phi(z_i)=0\) for all \(i \geq 0\). Therefore, we have the following relation for each \(i\geq m\): $$ c_{m}z_{i}+c_{m-1}z_{i-1}+\cdots+c_0z_{i-m}=0 $$ Now, let's substitute \(v=\left(-c_{0}, -c_{1}, \ldots, -c_{m-1}\right)\) and compute \(vA\). By definition, the \(j\)-th entry of the product \(vA\) is: $ (vA)_j = v \cdot a_j = -c_0z_j - c_1z_{j-1} - \cdots - c_{m-1}z_{j-m+1} $ We observe that for \(j=m, \ldots, 2m-1\), we have \((vA)_j = z_j - z_{j-1} - \cdots - z_{j-m+1} = w_j\). Thus, we have shown that \(vA=w\).

Step 2: Prove the uniqueness of the solution

Assume that there is another vector \(v' \in F^{1 \times m}\) such that \(v'A=w\). Then, \(v'-v\) should also satisfy \((v'-v)A=0\). Let's denote \(v'-v = u\). Then, the \(j\)-th entry of the product \(uA\) is: $ (uA)_j = u \cdot a_j = u_0z_j + u_1z_{j-1} + \cdots + u_{m-1}z_{j-m+1} $ As \((uA)_j = 0\) for \(j=m, \ldots, 2m-1\), we know that \(u\) and the minimal polynomial \(\phi\) satisfy the same relations for \(z_j\). However, a minimal polynomial is defined as the unique monic polynomial of the smallest degree that satisfies the relation. Therefore, \(u\) should be a multiple of \(\phi\), but noting that the leading coefficient of \(\phi\) is 1 (i.e., \(c_m=1\)), we can deduce that \(u\) should be equal to \(\phi\) up to a constant. As \(\phi\) has its coefficients negated in \(v\), we can conclude that \(u=-\phi\) and thus \(v'-v=-\phi\). However, this contradicts our assumption that \(v'\) is another solution to \(v'A=w\) since we already deduced that \(v\) is the unique solution with coefficients negated from \(\phi\). Therefore, \(v\) is the unique solution to the equation \(vA=w\). By following the steps above, we have shown that \(v=\left(-c_{0}, \ldots,-c_{m-1}\right) \in F^{1\times m}\) is the unique solution to the equation \(vA=w\).

Key Concepts

Minimal Polynomial

The concept of a minimal polynomial is central to the study of sequences and linear algebra. It is the unique monic polynomial of least degree that annihilates a sequence or a linear operator. In simpler terms, a minimal polynomial for a sequence is a polynomial such that each term of the sequence satisfies the polynomial equation, and there's no such polynomial of lower degree that can do that.

For instance, if we have a sequence \( \Psi = \{z_i\} \), where each subsequent element is defined by a linear combination of the previous terms, the minimal polynomial \( \phi \) captures this relationship succinctly. If we find such a polynomial with coefficients \( c_0, c_1, \ldots, c_m \) where \( c_m = 1 \) (indicating it's monic), then it provides a powerful tool for not just understanding, but also generating the sequence using its previous terms.

When we say that a sequence is 'linearly generated', we mean that past a certain point, every term can be generated by a linear combination of a fixed number of previous terms, with the coefficients of this combination being constants. The minimal polynomial reflects this by providing the smallest set of constants that can be used to generate the sequence indefinitely.

Sequence Characterization

Characterizing a sequence means describing its properties in a way that uniquely identifies it. Of significant importance in this context is how terms of the sequence are related to each other. For a linearly generated sequence, this relationship can be expressed using its minimal polynomial.

By examining the terms of the sequence and the coefficients of its minimal polynomial, we can find patterns that persist throughout the sequence. These patterns are essential for both sequence characterization and for predicting future terms of the sequence. For example, the relation \( c_{m}z_{i}+c_{m-1}z_{i-1}+\cdots+c_0z_{i-m}=0 \) derived from the minimal polynomial tells us how to compute any new term in the sequence by a specific linear combination of m previous terms.

Once understood, this characterization is not just a description, but a tool we can utilize. With it, we can create a matrix structure that encapsulates the pattern of the sequence. Using this matrix with the minimal polynomial, we can simplify the process of computing terms or even solve for unknown coefficients by using matrix operations.

Matrix Multiplication

Matrix multiplication is an operation that takes two matrices and produces another matrix. It's not just a fundamental operation in linear algebra, but a powerful tool to solve equations like those we see with linearly generated sequences.

In our exercise, we see matrix multiplication in action as we multiply the vector \( v \) with the matrix \( A \) to get the vector \( w \) - a process that uses the terms of our linearly generated sequence. To understand the computation, we need to match each term \( v_i \) of the vector \( v \) with each column \( a_j \) of the matrix \( A \) and sum their products.

In the case of our example, matrix multiplication encapsulates the behaviour dictated by the minimal polynomial and creates a compact equation, \( vA = w \), where \( v \) is the unique solution. This equation asserts that the relationship expressed by the sequence's minimal polynomial, along with the terms of the sequence itself, determine the vector \( w \) in a very precise manner.

This process not only proves the power and utility of matrices in dealing with sequences and systems of equations but also demonstrates the close interplay between different areas of algebra when it comes to understanding and characterizing mathematical structures.