Type a command or search the notes. Tab cycles completions. Up and down recall history when blank. Escape dismisses, then leaves the prompt.

[core/

]

Core Concept Map Language

Methods from Spectral applications

All methods from Spectral applications.

Back to Spectral applications

[METHOD]

Compute a matrix power by diagonalisation

To compute $A^k x_0$ for a non-negative integer $k$ when $A$ is diagonalisable:

choose an eigenbasis and form $P$ from its eigenvectors;
form the diagonal matrix $D$ from the corresponding eigenvalues;
write $A=PDP^{-1}$ ;
replace $A^k$ by $PD^kP^{-1}$ , so the powers are taken only on the diagonal;
apply the result to $x_0$ .

Equivalently, if $x_0=c_1v_1+\cdots+c_nv_n$ in the eigenbasis, then

A^kx_0 =c_1\lambda_1^kv_1+\cdots+c_n\lambda_n^kv_n.

This form shows each mode evolving on its own.

[1]:Diagonalisation
[2]:Powers after diagonalisation
[3]:Evolve a Markov system by diagonalisation
[4]:Markov systems
[5]:Diagonalisation of a stress matrix
[6]:Long-term behaviour of the modes
2 more

This page builds on Diagonalisation, Eigenvalues and eigenvectors and Stability of systems (Complex numbers).

It leads to Seen Questions and Markov systems.

Part of the thread Roots, modes and evolution, in 7 places.

A linear evolution splits into modes $r^n$ or $e^{rt}$ , and the roots $r$ decide growth, decay and oscillation.

THLong-term behaviour of the modesmatrix powersALG 4 / Here

On this page

Compute a matrix power by diagonalisation