|
Size: 1486
Comment: Exponentiation
|
Size: 1780
Comment: Rewrite
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 11: | Line 11: |
| == Introduction == | == Description == |
| Line 13: | Line 13: |
| A [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]] has many useful properties. A [[LinearAlgebra/MatrixProperties#Diagonalizability|diagonalizable matrix]] is a ''square'' matrix that can be factored into one. A square matrix that cannot be factored like this is called '''defective'''. | A '''diagonal matrix''' is a diagonal line of numbers in a square matrix of zeros. Such a matrix has many useful properties. * Its columns are its [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]] * The numbers in the diagonal are the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] * The [[LinearAlgebra/Determinant|determinant]] is the project of the numbers in the diagonal A matrix is '''diagonalizable''' if it can be factored into a diagonal matrix. Only square matrices can be diagonalizable. A square matrix that still cannot be factored as such is '''defective'''. |
| Line 19: | Line 26: |
| == Definition == | == Process == |
| Line 27: | Line 34: |
| ---- | |
| Line 29: | Line 37: |
| === Properties === | == Usage == |
Diagonalization
Diagonalization is an alternative decomposition of square matrices.
Contents
Description
A diagonal matrix is a diagonal line of numbers in a square matrix of zeros.
Such a matrix has many useful properties.
Its columns are its eigenvectors
The numbers in the diagonal are the eigenvalues
The determinant is the project of the numbers in the diagonal
A matrix is diagonalizable if it can be factored into a diagonal matrix. Only square matrices can be diagonalizable. A square matrix that still cannot be factored as such is defective.
Process
Given a matrix A, notate the matrix of its eigenvectors as S. A diagonalizable matrix can be factored as A = SΛS-1.
Λ will be a diagonal matrix with the eigenvalues of A in the diagonal.
In other words, A can be rewritten as a eigennormalized (i.e. transformed by S) then un-eigennormalized (i.e. transformed by S-1) diagonal matrix Λ.
Usage
Diagonalization offers clean solutions to mathematical models.
A2 = SΛ2S-1, and more generally AK = SΛKS-1.
Similarly, eA = SeΛS-1. Note that eΛ is a diagonal matrix with e to the power of the eigenvalues of A in the diagonal.