Diagonalization and Similarity

Suppose you need to know what a matrix does when applied 50 times in a row — a Markov chain's transition probabilities after 50 steps, or a discretized differential equation propagated 50 periods forward. Multiplying the matrix by itself 50 times by hand is slow and the intermediate numbers get ugly fast. Diagonalization is a shortcut: if you can find the right coordinate system, the matrix's repeated action becomes nothing more than raising a handful of plain numbers to the 50th power.

An analogy: describing wind along its own direction

If wind blows diagonally across a field, describing its speed using "north" and "east" components is awkward — both numbers change every time you rotate your reference frame, and neither one alone tells you the true speed. But if you instead measure along the wind's own direction, one number captures everything, and it stays simple no matter how many times you re-measure. Diagonalization does the matrix equivalent: it finds the special directions (eigenvectors) along which the matrix acts as pure scaling, no rotation or shearing. Described along those directions, a matrix's repeated action is just repeated multiplication by a single scaling number per direction.

The math, one symbol at a time

A square matrix $A$ is diagonalizable if it can be written as

$$A = PDP^{-1},$$

where $D$ is a diagonal matrix holding $A$'s eigenvalues $\lambda_1, \ldots, \lambda_n$ down the diagonal, and $P$ is the matrix whose columns are the corresponding eigenvectors. In words: $A$ equals "change into eigenvector coordinates" ($P^{-1}$), "scale each coordinate by its eigenvalue" ($D$), "change back" ($P$). Two matrices related this way — $A$ and $D$ — are called similar: they represent the same underlying linear transformation, just viewed in different coordinate systems, and similar matrices always share the same eigenvalues, determinant, and trace.

The payoff is powers: $A^n = PD^nP^{-1}$, and since $D$ is diagonal, $D^n$ just raises each diagonal entry to the $n$-th power — no matrix multiplication required for the hard part.

Worked example 1: computing a matrix cube

Let $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$. Its eigenvalues solve $(2-\lambda)^2 - 1 = 0$, giving $\lambda_1 = 3$, $\lambda_2 = 1$, with eigenvectors $(1,1)$ and $(1,-1)$. So $P = \begin{pmatrix}1&1\\1&-1\end{pmatrix}$, $D = \begin{pmatrix}3&0\\0&1\end{pmatrix}$, and $P^{-1} = \frac{1}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$. Then $A^3 = PD^3P^{-1}$ with $D^3 = \begin{pmatrix}27&0\\0&1\end{pmatrix}$. Multiplying out gives $A^3 = \begin{pmatrix}14&13\\13&14\end{pmatrix}$ — the same answer direct multiplication gives, but reached by cubing two numbers (27 and 1) instead of multiplying $2\times2$ matrices twice.

Worked example 2: a two-state Markov chain after many steps

A trading signal flips between "on" (state 1) and "off" (state 0) with transition matrix $A = \begin{pmatrix}0.9 & 0.2 \\ 0.1 & 0.8\end{pmatrix}$ (columns: probability of going from that state to each row's state). Its eigenvalues are $\lambda_1 = 1$ and $\lambda_2 = 0.7$. Diagonalizing and raising to a high power $n$, the $\lambda_2^n = 0.7^n$ term shrinks toward zero as $n$ grows, leaving only the $\lambda_1 = 1$ contribution — this is exactly why Markov chains settle into a steady-state distribution: every eigenvalue below 1 dies out under repeated multiplication, and diagonalization makes that decay visible as ordinary exponential decay of a single number.

Drag the matrix entries and watch the ellipse the unit circle maps to; the two directions that stay fixed (only stretched, not rotated) are the eigenvectors — exactly the axes diagonalization measures along.

What this means in practice

Quants lean on diagonalization constantly: computing long-horizon transition probabilities in Markov models, solving linear systems of differential equations (interest-rate or population dynamics), and understanding principal component analysis, where diagonalizing a covariance matrix reveals the independent, uncorrelated risk directions in a portfolio. It's also the backbone of the The Matrix Exponential, which underlies continuous-time Markov chains and many pricing PDEs.