Eigendecomposition of a matrix

Every invertible matrix

M

can be written as:

M = Q D Q^{- 1}

(39)

where:

$D$ is a diagonal matrix containing the eigenvalues of $M$
columns of $Q$ are eigenvectors of $M$

Note therefore that this decomposition is unique up to swapping the order of eigenvectors. We could fix a canonical form by sorting eigenvectors from smallest to largest in the case of a real number.

Intuitively, Note that this is just the change of basis formula, and so:

$Q^{- 1}$ changes basis to align to the eigenvectors
$D$ multiplies eigenvectors simply by eigenvalues
$Q$ changes back to the original basis

 Ancestors (7)

Eigenvalues and eigenvectors
Matrix
Linear algebra
Algebra
Area of mathematics
Mathematics
 Home

 Incoming links (2)

Eigendecomposition of a real symmetric matrix
Sylvester's law of inertia

 Synonyms (1)

Eigendecomposition