Diagonalizing a matrix involves transforming it into a diagonal matrix using a similarity transformation, which requires finding the matrix's eigenvalues and eigenvectors.
The Process of Diagonalization
A square matrix A is diagonalizable if and only if there exists an invertible matrix P such that P⁻¹AP is a diagonal matrix, often denoted as D. The matrix D contains the eigenvalues of A on its main diagonal, and the columns of the matrix P are the corresponding eigenvectors.
Here are the key steps to diagonalize a matrix, based on finding the necessary matrices P and D:
- Compute the eigenvalues of A.
Find the roots of the characteristic equation det(A - λI) = 0, where λ represents the eigenvalues and I is the identity matrix. These roots are the eigenvalues of A. - Check that no eigenvalue is defective.
For a matrix to be diagonalizable, it must have a complete set of linearly independent eigenvectors. This occurs if and only if, for every eigenvalue, its algebraic multiplicity (the number of times it appears as a root of the characteristic equation) equals its geometric multiplicity (the dimension of the eigenspace corresponding to that eigenvalue). A defective eigenvalue has an algebraic multiplicity greater than its geometric multiplicity, preventing diagonalization via this method. - For each eigenvalue, find as many linearly independent eigenvectors as you can.
The number of linearly independent eigenvectors for an eigenvalue equals its geometric multiplicity. To find the eigenvectors corresponding to an eigenvalue λ, solve the system of linear equations (A - λI)v = 0, where v is the eigenvector. - Adjoin all the eigenvectors so as to form a full-rank matrix P.
Collect all the linearly independent eigenvectors found in the previous step. These eigenvectors will form the columns of the matrix P. Ensure that the number of eigenvectors equals the dimension of the matrix A (e.g., for an nxn matrix, you need n linearly independent eigenvectors). If you have n linearly independent eigenvectors, the matrix P formed by these vectors as columns will be full-rank and thus invertible. - Construct the diagonal matrix D.
Create a diagonal matrix D where the diagonal entries are the eigenvalues corresponding to the eigenvectors in P, in the same order. If the i-th column of P is the eigenvector for eigenvalue λᵢ, then the i-th diagonal entry of D is λᵢ.
If these steps are successfully completed, you have found the matrices P and D such that A = PDP⁻¹, or equivalently, D = P⁻¹AP. This transformation simplifies many calculations involving A, particularly matrix powers and exponentials.
You can find more details on this process at the Statlect website.
Why This Works: The Diagonalization Equation
The relationship A = PDP⁻¹ is the core of diagonalization. P can be viewed as a change-of-basis matrix. The eigenvectors in P form a basis where the linear transformation represented by A acts simply as scaling along the directions of the eigenvectors. The matrix D represents this scaling in the eigenvector basis. Multiplying by P⁻¹ (changing from the standard basis to the eigenvector basis), applying the simple scaling D, and then multiplying by P (changing back to the standard basis) is equivalent to applying the original transformation A.
Diagonalization is a powerful tool in linear algebra with applications in differential equations, quantum mechanics, and data analysis (like Principal Component Analysis).
