Diagonalization

Diagonalization is an alternative decomposition of square matrices.

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 A^K = SΛ^KS^-1.

Similarly, e^A = Se^ΛS^-1. Note that e^Λ is a diagonal matrix with e to the power of the eigenvalues of A in the diagonal.

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