Incoming links: Invertible matrix
Name origin: likely because it "determines" if a matrix is invertible or not, as a matrix is invertible iff determinant is not zero.
Every invertible matrix can be written as:where: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.
- is a diagonal matrix containing the eigenvalues of
- columns of are eigenvectors of
Intuitively, Note that this is just the change of basis formula, and so:
- changes basis to align to the eigenvectors
- multiplies eigenvectors simply by eigenvalues
- changes back to the original basis
Some specific examples:
The set of all invertible matrices forms a group: the general linear group with matrix multiplication. Non-invertible matrices don't form a group due to the lack of inverse.
When it exists, which is not for all matrices, only invertible matrix, the inverse is denoted: