Diagonalisation Applications

Level 1 - Math I (Physics) topic page in Linear Maps.

Principle

Diagonalisation makes repeated linear operations simple. Instead of applying a coupled matrix many times, change to eigenvector coordinates, scale independently, and change back.

Power through diagonalisation

\[A^n=PD^nP^{-1}\]

This appears in discrete time evolution, coupled oscillations, normal modes, and linear stability calculations.

Notation

\(A^n\)

matrix A applied n times

\(D^n\)

diagonal matrix with diagonal entries raised to nth powers

\(\mathbf x_0\)

initial state

\(\mathbf x_n\)

state after n updates

\(P^{-1}\mathbf x\)

coordinates of x in the eigenvector basis

Eigenvalues control the long-term size of components. Eigenvectors tell which independent directions those components occupy.

Method

Step 1: Diagonalise the matrix

Find \(A=PDP^{-1}\).

Step 2: Move the initial vector into eigenvector coordinates

Compute \(P^{-1}\\mathbf x_0\). These coordinates are mode amplitudes.

Step 3: Evolve each mode independently

Use \(D^n\), which is easy because \(D\) is diagonal.

Rules

Diagonal powers

\[D=\operatorname{diag}(\lambda_1,\lambda_2,\cdots,\lambda_n)\quad\Rightarrow\quad D^k=\operatorname{diag}(\lambda_1^k,\lambda_2^k,\cdots,\lambda_n^k)\]

Repeated update

\[\mathbf x_n=A^n\mathbf x_0=PD^nP^{-1}\mathbf x_0\]

Modes with \(\\lvert\\lambda\\rvert\\lt1\) decay under repeated updates. Modes with \(\\lvert\\lambda\\rvert\\gt1\) grow. Modes with negative eigenvalues alternate sign.

Examples

Question

\[D=\operatorname{diag}(2,5)\]

what is \(D^3\)?

Answer

Raise each diagonal entry to the third power:

\[D^3=\operatorname{diag}(8,125)\]

Checks

Use diagonalisation only after checking that enough independent eigenvectors exist.
Powers of \(D\) are easy because only diagonal entries are powered.
Long-term behaviour is controlled by eigenvalue magnitudes.
Convert back to original coordinates when the final answer needs original variables.

Diagonalisation