Cophase Roots: Definition and Polynomial Representation

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome back to our course.

In the previous module, we introduced the algebraic machinery needed to analyze the paper's core problem, specifically symmetric functions and Schur polynomials. Now, we'll begin to see how these tools are applied to find the solution.

This lesson is the first in our fifth module, "The Solution: Cophase Roots and Schur Positivity." We will focus on understanding the configuration of characteristic roots that the paper identifies as optimal.

Lesson Goal: By the end of this lesson, you will be able to define 'cophase roots' and explain why a characteristic polynomial of the form $(z - r)^{n}$ is the mathematical representation of this configuration.

1. Defining "Cophase Roots"

The paper's main result identifies a specific arrangement of characteristic roots that maximizes the amplitude $∣ x_{t} ∣$ . The authors give this arrangement a specific name. Let's start by looking at the original text to find out what it is and how it's described.

Amplitude maximization in stable systems, Schur positivity, ...

Please read the abstract of the paper 'Amplitude maximization in stable systems, Schur positivity, ...'. This will give us the authors' own description of their key finding and introduce the terminology we'll be using.

Read the abstract on the first page. Focus on the sentence that describes where the maximum is attained and what this implies for the characteristic polynomial.

As you read, you'll notice two key phrases:

"...the maximum is attained with coinciding roots on the boundary circle..."
"...the optimality of the cophase root configuration..."

These phrases describe the same concept. Let's break it down:

Boundary Circle: In our problem, the characteristic roots $z_{1}, \dots, z_{n}$ are constrained to lie within a disc of radius $r$ , i.e., $∣ z_{k} ∣ \leq r$ . The "boundary circle" is the circle where this inequality becomes an equality: $∣ z_{k} ∣ = r$ .
Coinciding Roots: This simply means all the roots are identical: $z_{1} = z_{2} = \dots = z_{n}$ .
Cophase: In the complex plane, a number $z = ∣ z ∣ e^{i ϕ}$ has a magnitude $∣ z ∣$ and a phase (or angle) $ϕ$ . "Co-phase" means all roots share the same phase. If they also have the same magnitude (as they do here, being on the boundary circle), they must be the same complex number.

Therefore, a cophase root configuration on the boundary means all characteristic roots of the difference equation are identical and have the maximum possible magnitude, $r$ . We can write this as:
$z_{1} = z_{2} = \dots = z_{n} = z_{0}$
where $z_{0}$ is a complex number with $∣ z_{0} ∣ = r$ . This means $z_{0}$ can be expressed as $r e^{i θ}$ for some angle $θ \in [0, 2 π)$ .

2. The Characteristic Polynomial for Cophase Roots

Now, let's connect this root configuration to the characteristic polynomial.

Recall that a monic polynomial $f (z)$ of degree $n$ with roots $z_{1}, \dots, z_{n}$ can be written in factored form as:
$f (z) = (z - z_{1}) (z - z_{2}) \dots (z - z_{n})$

If we have a cophase root configuration where all roots are equal to $z_{0} = r e^{i θ}$ , the characteristic polynomial becomes:
$f (z) = (z - r e^{i θ}) (z - r e^{i θ}) \dots (z - r e^{i θ}) = (z - r e^{i θ})^{n}$

The abstract you read mentions a specific polynomial: $(z - r)^{n}$ . This is the specific instance of the cophase root polynomial where the common phase is $θ = 0$ , making the root a positive real number, $r$ .

The paper's main theorem confirms this and shows that the magnitude of the maximum amplitude $∣ x_{t} ∣$ is independent of the phase $θ$ . Let's look at the formal statement.

Amplitude maximization in stable systems, Schur positivity, ...

Now, let's examine Theorem 1 in the paper. It formally states the main result and uses the term 'cophase roots' explicitly.

Find Theorem 1 in Section 1. Pay attention to equation (5) and the sentence immediately following it. Notice the form of the characteristic polynomial that achieves the maximum.

Theorem 1 states that the maximum amplitude is achieved by a polynomial of the form $\prod_{k = 1}^{n} (z - r_{k} e^{i θ})$ . In the simpler case we've been considering, where all roots are constrained by the same radius $r$ , this becomes $(z - r e^{i θ})^{n}$ . The theorem then concludes: "Thus, $∣ x_{t} ∣$ is maximized with cophase roots."

This confirms that the polynomial $(z - r)^{n}$ is the canonical representation of the optimal cophase root configuration.

3. Consequence of Repeated Roots

Why is this configuration of roots so effective at maximizing amplitude? The answer lies in the structure of the solution $x_{t}$ that arises from a characteristic polynomial with a highly repeated root.

Your background in physics and simulation likely involved solving differential equations. You may recall that when the characteristic equation of a linear ODE has repeated roots, the solution includes polynomial terms multiplied by the exponential term (e.g., $t e^{λ t}$ ). A similar phenomenon occurs with difference equations.

To see this clearly, let's consult a standard text on the topic.

Section 4 Linear difference equations

This resource provides a concise explanation of how to find the general solution for a linear difference equation, including the special case of repeated roots.

Please read the text in Section 4.1. Focus on the two paragraphs starting with 'For a general characteristic equation...' and 'If we have a repeated root...'. Then, look at Example 4.2, which shows a concrete application of the repeated root case.

As the resource explains, if a characteristic root $λ_{1}$ is repeated $r$ times, the corresponding part of the general solution is not just $C_{1} λ_{1}^{t}$ , but rather:
$x_{t} = (D_{0} + D_{1} t + \dots + D_{r - 1} t^{r - 1}) λ_{1}^{t}$
where $D_{k}$ are constants determined by the initial conditions.

Now, let's apply this to our cophase root configuration. The characteristic polynomial is $(z - r)^{n}$ , which means we have a single root, $z_{1} = r$ , repeated $n$ times.

Based on the rule for repeated roots, the general solution to the difference equation must have the form:
$x_{t} = (c_{0} + c_{1} t + c_{2} t^{2} + \dots + c_{n - 1} t^{n - 1}) r^{t}$
Here, $P (t) = (c_{0} + c_{1} t + \dots + c_{n - 1} t^{n - 1})$ is a polynomial of degree $n - 1$ .

This structure is the key. The term $r^{t}$ controls the exponential growth or decay. The polynomial term $P (t)$ provides an additional source of growth. For a fixed time $t$ , by choosing the initial conditions $(x_{0}, \dots, x_{n - 1})$ appropriately, we can manipulate the coefficients $c_{k}$ of the polynomial to maximize its value, and thus maximize the overall amplitude $∣ x_{t} ∣$ . The presence of terms up to $t^{n - 1}$ is a direct consequence of all $n$ roots "collapsing" to a single point, and this is what enables the maximal transient amplification.

Conclusion

In this lesson, we have defined the central concept of "cophase roots" and understood its mathematical representation.

Key Takeaways:

Cophase Roots: This refers to a configuration where all characteristic roots of the difference equation are identical and lie on the boundary of the allowed domain. For a disc of radius $r$ , this means $z_{1} = \dots = z_{n} = r e^{i θ}$ .
Characteristic Polynomial: The polynomial corresponding to this configuration is $(z - r e^{i θ})^{n}$ . The paper often uses the simplified case $(z - r)^{n}$ , which corresponds to a phase of $θ = 0$ .
Solution Structure: A characteristic polynomial with a single root repeated $n$ times leads to a general solution of the form $x_{t} = P (t) r^{t}$ , where $P (t)$ is a polynomial of degree $n - 1$ . This polynomial prefactor is crucial for maximizing the amplitude.

Next Lesson Preview:

We have now identified the what—the optimal root configuration is cophase. In the next lesson, we will begin to explore the why. We will connect the concept of Schur positivity, which we studied in the previous module, to the argument that proves this cophase root configuration is indeed the one that maximizes the amplitude.