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

Diagonalization is a decomposition that turns a matrix into a diagonal matrix, with some change of basis matrices to the left and right of it. The idea follows from eigenvectors:

• Recall that Ax = λx.

• It follows that AS = SΛ where...

  • S is the eigenbasis composed of all eigenvectors x.

  • Λ is the diagonal matrix where the numbers in the diagonal are the eigenvalues.

Recall that only square matrices have eigenvectors, and that a matrix of size n x n either has n unique eigenpairs or is defective. These restrictions apply to diagonalization; a matrix that cannot be diagonalized is defective.

Procedure

The procedure for diagonalizing A is:

• solving for eigenvalues and eigenvectors

• constructing S and Λ as described above

  • Note that order of eigenpairs does not matter, as long as the orders in S and Λ match.

• inverting S

In the case of a symmetric matrix A, the eigenbasis can be transposed instead of calculating the inverse. To indicate this, the eigenbasis is generally notated as Q instead, as in A = QΛQ^T.

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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