Diagonalisation

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

Principle

Diagonalisation rewrites a matrix using a basis of eigenvectors. In that basis, the linear map acts by independent scaling along coordinate directions.

Diagonalisation

\[A=PDP^{-1}\]

This is powerful in physics because it separates coupled linear behaviour into independent modes.

Notation

\(A\)

matrix to diagonalise

\(P\)

matrix whose columns are eigenvectors

\(D\)

diagonal matrix of eigenvalues

\(P^{-1}\)

change from standard coordinates to eigenvector coordinates

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

A is diagonalised by P

Diagonalisation is possible when there are enough linearly independent eigenvectors to form a basis.

Method

Step 1: Find eigenvalues

Solve \(\\det(A-\\lambda I)=0\).

Step 2: Find independent eigenvectors

For each eigenvalue, solve \((A-\\lambda I)\\mathbf v=\\mathbf0\). Collect enough independent eigenvectors to form a basis.

Step 3: Build \(P\) and \(D\)

Put eigenvectors as columns of \(P\). Put the matching eigenvalues in the same order down the diagonal of \(D\).

Rules

Column matching

\[A\mathbf v_i=\lambda_i\mathbf v_i\]

Matrix form

\[AP=PD\]

Diagonal form

\[D=P^{-1}AP\]

If \(A\) has \(n\) distinct eigenvalues, then an \(n\\times n\) matrix is diagonalizable. Repeated eigenvalues require checking whether enough independent eigenvectors exist.

Examples

Question

Diagonalise

\[A=\begin{pmatrix}2&0\\0&5\end{pmatrix}\]

Answer

The standard basis vectors are eigenvectors and the matrix is already diagonal. We can take

\[P=I\]

and

\[D=A\]

Checks

The columns of \(P\) must be independent eigenvectors.
The order of columns in \(P\) must match the order of eigenvalues in \(D\).
A matrix with repeated eigenvalues may fail to diagonalise.
Verify either \(AP=PD\) or \(A=PDP^{-1}\).

Eigenspaces

Diagonalisation Applications