Powers and Polynomials Under Similarity

Similarity and Repeated Multiplication

Similar matrices represent the same linear operator in different bases. A change of basis can make a matrix easier to work with, but it is useful only if calculations performed in the simpler representation can be translated back correctly. Powers and polynomials are two important cases where this translation has a precise rule.

Throughout, let \(\mathbf{A},\mathbf{B},\mathbf{P}\in\mathbb{R}^{n\times n}\), with \(\mathbf{P}\) invertible. The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are similar when \(\mathbf{B}=\mathbf{P}^{-1}\mathbf{A}\mathbf{P}\). Matrix multiplication is associative, but in general it is not commutative: the order of factors matters. In particular, the factors \(\mathbf{P}\) and \(\mathbf{P}^{-1}\) can be cancelled when they are adjacent, but they cannot be moved past \(\mathbf{A}\) at will.

This definition includes constant polynomials: if \(q(t)=c_0\), then \(q(\mathbf{A})=c_0\mathbf{I}_n\). The zero polynomial gives the zero matrix. Polynomial evaluation uses sums and powers of one matrix; it does not mean that arbitrary matrix factors can be rearranged.

The Identity for Powers

Proof. We use induction on \(k\). For \(k=0\), the definition of the zeroth power and the inverse identity give \(\mathbf{B}^0=\mathbf{I}_n=\mathbf{P}^{-1}\mathbf{I}_n\mathbf{P}=\mathbf{P}^{-1}\mathbf{A}^0\mathbf{P}\). Now suppose the formula holds for some \(k\ge0\). Using the definition \(\mathbf{B}=\mathbf{P}^{-1}\mathbf{A}\mathbf{P}\), associativity, and \(\mathbf{P}\mathbf{P}^{-1}=\mathbf{I}_n\), we obtain \(\mathbf{B}^{k+1}=\mathbf{B}^k\mathbf{B}=(\mathbf{P}^{-1}\mathbf{A}^k\mathbf{P})(\mathbf{P}^{-1}\mathbf{A}\mathbf{P})=\mathbf{P}^{-1}\mathbf{A}^k(\mathbf{P}\mathbf{P}^{-1})\mathbf{A}\mathbf{P}=\mathbf{P}^{-1}\mathbf{A}^{k+1}\mathbf{P}\). Thus the formula holds at \(k=0\), and whenever it holds at \(k\) it holds at \(k+1\); it follows for every integer \(k\ge0\). \(\square\)

The proof works because each adjacent pair \(\mathbf{P}\mathbf{P}^{-1}\) cancels as the product is extended by one more factor. The matrices are square and \(\mathbf{P}\) is invertible, which ensures that both \(\mathbf{P}^{-1}\) and the identities involving \(\mathbf{I}_n\) are defined. No diagonalization assumption is involved.

The Identity for Polynomials

The power formula extends to any polynomial because evaluating a polynomial means taking a finite linear combination of powers. The coefficients are scalars, so they multiply the matrices without changing their order.

Proof. By the definition of polynomial evaluation and the power theorem, including its case \(j=0\), \(q(\mathbf{B})=\sum_{j=0}^{m}c_j\mathbf{B}^j=\sum_{j=0}^{m}c_j(\mathbf{P}^{-1}\mathbf{A}^j\mathbf{P})\). For each \(j\), the scalar \(c_j\) can be placed between the matrices as \(c_j(\mathbf{P}^{-1}\mathbf{A}^j\mathbf{P})=\mathbf{P}^{-1}(c_j\mathbf{A}^j)\mathbf{P}\). Distributing the finite sums and factoring the common matrices on the left and right therefore gives \(q(\mathbf{B})=\mathbf{P}^{-1}(\sum_{j=0}^{m}c_j\mathbf{A}^j)\mathbf{P}=\mathbf{P}^{-1}q(\mathbf{A})\mathbf{P}\). If \(q\) is the zero polynomial or has degree zero, the same calculation applies; in particular, \(\mathbf{P}^{-1}(c_0\mathbf{I}_n)\mathbf{P}=c_0\mathbf{I}_n\). \(\square\)

This theorem gives a two-way computational option. One may first evaluate \(q\) at \(\mathbf{A}\) and then transfer the result, or first form the similar matrix \(\mathbf{B}\), evaluate \(q\) there, and obtain the same answer. The choice is useful when one matrix has a simpler structure than the other.

Simplifying a Large Power

A particularly effective application is to choose a similar matrix whose powers are easy to compute. The next example uses a diagonal matrix. The relevant calculation is just repeated multiplication of diagonal entries; no claim about eigenvalues is needed.

Common Pitfall: Reordering the Factors

The identity does not permit moving \(\mathbf{P}\) or \(\mathbf{P}^{-1}\) through \(\mathbf{A}\). A common incorrect step is to claim \((\mathbf{P}^{-1}\mathbf{A}\mathbf{P})^2=\mathbf{P}^{-2}\mathbf{A}^2\mathbf{P}^2\). That expression is not the result of the power theorem: it wrongly combines factors on opposite sides of \(\mathbf{A}\). In the first worked example, that incorrect expression would use \(\mathbf{P}^{-2}=\begin{bmatrix}1&0\\0&\tfrac14\end{bmatrix}\) and \(\mathbf{P}^2=\begin{bmatrix}1&0\\0&4\end{bmatrix}\). It would therefore give

which is not \(\mathbf{B}^2=\begin{bmatrix}4&10\\0&9\end{bmatrix}\). The correct expansion keeps the factors in their original order: \((\mathbf{P}^{-1}\mathbf{A}\mathbf{P})(\mathbf{P}^{-1}\mathbf{A}\mathbf{P})=\mathbf{P}^{-1}\mathbf{A}(\mathbf{P}\mathbf{P}^{-1})\mathbf{A}\mathbf{P}=\mathbf{P}^{-1}\mathbf{A}^2\mathbf{P}\). Only the adjacent inverse pair cancels.

When These Identities Are Useful

Similarity lets us move a calculation between two representations without changing its mathematical content. If \(\mathbf{B}\) is simpler to raise to a power or substitute into a polynomial, compute there and use the identity to relate the result to the corresponding expression in \(\mathbf{A}\). For example, a diagonal matrix makes powers particularly direct, while a matrix with many zero entries can make a polynomial calculation shorter.

The formulas apply to every polynomial and every nonnegative integer power, whether or not the matrices have any additional structure. They do not say that \(\mathbf{A}\) and \(\mathbf{B}\) are equal, nor do they allow arbitrary products involving other matrices to be rearranged. The reliable method is to keep the conjugating matrices at the two ends, preserve the order of all factors in the middle, and cancel only adjacent \(\mathbf{P}\mathbf{P}^{-1}\) pairs.