Diagonalisation

Diagonalisation

Basic Concepts

• Diagonalisation is the process used to take a square matrix and convert it into a diagonal matrix, by means of a similarity transformation.
• A matrix is said to be diagonalisable if it can be written in the form PDP^-1, where P is a non-singular matrix and D is a diagonal matrix.
• In the above form, P^-1 is the inverse of the matrix P and D is the diagonal matrix.
• If the diagonal matrix D consists of eigenvalues λ1, λ2, …, λn, and corresponding eigenvectors form the columns of P.
• Diagonal matrices hugely simplify matrix operations such as exponentiation.

Procedure for Diagonalisation

• To diagonalise a matrix A, first find the eigenvalues of A.
• After finding the eigenvalues, find the associated eigenvectors. Each eigenvalue will have a corresponding eigenvector.
• Form matrix P with the eigenvectors as its columns, and construct the diagonal matrix D using the eigenvalues.
• Finally, verify that A = PDP^-1 to ensure the matrix has been correctly diagonalised.

Properties and Uses of Diagonalisation

• Only square matrices can be diagonalised, but not all square matrices are diagonalisable.
• A matrix is diagonalisable if and only if it has n distinct eigenvalues, where n is the number of dimensions of the matrix.
• For matrices with repeated eigenvalues, the presence or absence of enough linearly independent eigenvectors determines diagonalisability.
• Diagonalisation is used in many areas of mathematics, computer science and engineering due to the simplifying property of diagonal matrices.

Diagonalisation and Powers of a Matrix

• One of the primary uses of diagonalisation involves calculating powers of a matrix.
• If A is a diagonalisable matrix and it can be expressed as A = PDP^-1, then any power of A, let’s say A^n, can be calculated as A^n = PD^nP^-1.
• This operation remains manageable even for large exponents because taking the power of a diagonal matrix D^n merely involves taking the power of each of its diagonal elements.