Question

Given a polynomial \(p(x)=r_{0}+r_{1} x+\) \(\cdots+r_{n} x^{n}\) and a square matrix \(A,\) the matrix \(p(A)=\) \(r_{0} I+r_{1} A+\cdots+r_{n} A^{n}\) is called the evaluation of \(p(x)\) at A. Let \(B=P^{-1} A P\). Show that \(p(B)=P^{-1} p(A) P\) for all polynomials \(p(x)\).

Short answer

The solution shows that for any polynomial, evaluating at a matrix B gives \( p(B) = P^{-1} p(A) P \).

Step-by-step solution

Understand the definitions

Let's first understand the definitions. The polynomial in matrix form is evaluated as \( p(A) = r_{0}I + r_{1}A + r_{2}A^2 + \ldots + r_{n}A^n \) where \( I \) is the identity matrix of the same size as matrix \( A \). Matrix \( B = P^{-1}AP \) is referred to as a similarity transformation of \( A \).

Express the polynomial for matrix B

The polynomial evaluated at matrix \( B \) is \( p(B) = r_{0}I + r_{1}B + r_{2}B^2 + \ldots + r_{n}B^n \). Using the definition of \( B \), replace \( B \) with \( P^{-1}AP \) in the polynomial expression.

Simplify each term of p(B)

Each term \( B^k \) for \( k \geq 1\) can be written as \( (P^{-1}A P)^k = P^{-1}A^k P \). This follows from repeated application of the property \( P^{-1}AP \cdot P^{-1}AP = P^{-1}A^2P \), and similar logic applies for higher powers. Thus, \( B^k = (P^{-1}A^kP) \).

Substitute and simplify the entire polynomial

Substituting \( B^k = P^{-1}A^kP \) back into the polynomial \( p(B) \), we have:\[p(B) = r_{0}I + r_{1}(P^{-1}A P) + r_{2}(P^{-1} A^2 P) + \ldots + r_{n}(P^{-1} A^n P)\]Factor \( P^{-1} \) and \( P \) out of the polynomial expression.\[p(B) = P^{-1}(r_{0}I + r_{1}A + r_{2}A^2 + \ldots + r_{n}A^n)P = P^{-1}p(A)P\]

Confirm and conclude

We have shown that each term in \( p(B) \) corresponds to a transformation involving \( P^{-1} \) and \( P \), and thus the entire polynomial evaluated at \( B \) can be expressed as \( P^{-1}p(A)P \), confirming that the relationship \( p(B) = P^{-1}p(A)P \) holds for any polynomial \( p(x) \).

Key concepts

similarity transformation

A similarity transformation is like giving a matrix a change of perspective. When we say that matrix \( B \) is the result of a similarity transformation of matrix \( A \), we're essentially saying \( B = P^{-1} A P \). This equation involves another matrix \( P \), which is invertible, meaning it can be flipped around to form \( P^{-1} \), its inverse. Similarity transformations help us to explore matrices from different angles without altering their essential properties.
Some of the benefits of similarity transformations include:

• Preserving eigenvalues: The eigenvalues of \( A \) and \( B \) are the same.
• Helping with computations: Sometimes, \( B \) can be a simpler form of \( A \), making complex calculations easier.
• Maintaining determinant: The determinants of \( A \) and \( B \) remain the same.

One of the key motivations for using similarity transformations is to simplify the problem we are dealing with, without losing the characteristics of the original matrix.

polynomials in linear algebra

Polynomials in linear algebra extend the familiar concept of polynomials to matrices. When you have a polynomial such as \( p(x) = r_{0} + r_{1}x + r_{2}x^2 + \cdots + r_{n}x^n \), plugging in a matrix \( A \) in place of the variable \( x \) gives us a polynomial in a matrix, \( p(A) \). This means replacing \( x^k \) with \( A^k \) in the expression. This is called evaluating the polynomial at matrix \( A \).
Here's how it works:

• \(I\) stands for the identity matrix. It behaves like the number 1 in regular algebra.
• The operation \( A^2 \) means the matrix \( A \) multiplied by itself.
• Each coefficient \( r_k \) is just a regular number that scales its corresponding matrices.

Evaluating a polynomial at a matrix is a way to construct new matrices that capture certain properties of \( A \). It can be particularly useful for solving more complex problems or proving key results in linear algebra.

matrix similarity

Matrix similarity ties closely with similarity transformations. Two matrices \( A \) and \( B \) are considered similar if one can be transformed into the other through a similarity transformation, meaning \( B = P^{-1} A P \). Similar matrices share many important characteristics:

• Eigenvalues: They have the same eigenvalues, which gives insight into their fundamental properties.
• Powers: Raising both matrices to any power will yield matrices with similar traits.
• Trace: The trace, which is the sum of the diagonal elements, remains consistent across similar matrices.

While they might look different at first glance, the essence of matrix similarity is that these matrices are just different views of the same original entity. This concept is profoundly useful because it allows us to study matrices by transforming them into simpler or more familiar forms. Understanding these underlying relationships helps in simplifying complex problems and calculations.