Uniqueness of Matrix Diagonalization

Matrix diagonalization, a fundamental concept in linear algebra, involves expressing a matrix in a simpler form where the matrix is diagonal. However, the diagonalization of a matrix is not unique, which has multiple implications and nuances. This article delves into the intricacies of matrix diagonalization, exploring when it is unique and when it is not.

Diagonalizable Matrices and Essential Concepts

A matrix (A) is diagonalizable if there exists an invertible matrix (P) and a diagonal matrix (D) such that (A PDP^{-1}). The diagonal entries of (D) are the eigenvalues of (A), and the columns of (P) are the corresponding eigenvectors. This relationship is foundational in understanding the properties of matrices and their applications in various scientific and engineering fields.

Multiplicity of Eigenvalues and Unique Diagonalization

The uniqueness of diagonalization can be influenced by the multiplicity of eigenvalues. If an eigenvalue has an algebraic multiplicity greater than its geometric multiplicity, multiple choices of basis eigenvectors can lead to different matrices (P). This is due to the fact that eigenvectors corresponding to eigenvalues with an algebraic multiplicity greater than one can be chosen freely up to a scalar multiple. For instance, if (v) is an eigenvector corresponding to an eigenvalue, then (cv) for any non-zero scalar (c) is also an eigenvector associated with the same eigenvalue.

Permutation of Eigenvectors and Scaling

Even when the eigenvectors are chosen uniquely, the order in which they are arranged in (P) can still lead to different diagonalization forms. Permuting the columns of (P) changes which eigenvector is associated with which diagonal entry in (D). Additionally, scaling the eigenvectors by a non-zero scalar does not change the diagonalization, as long as the matrix remains similar to the original diagonal form. This highlights the flexibility in choosing the specific representation of the diagonalization, even within the constraints of the eigenvalues and eigenvectors.

Matrix Diagonalization Beyond Uniqueness

Not all matrices can be diagonalized, and the question of whether a diagonalizable matrix can be unitarily (orthogonally) diagonalized is a pertinent one. Unitary diagonalization involves finding a unitary matrix (U) such that (A UDU^*), where (U^*) is the conjugate transpose of (U). A matrix can be unitarily diagonalized if and only if its eigenvectors can be chosen to be orthonormal, meaning they form an orthogonal basis for the vector space.

Non-Orthogonal Eigenvectors and Non-Unitary Diagonalization

To illustrate that a diagonalizable matrix can be unitarily diagonalized only if its eigenvectors are orthonormal, consider a simple example in (R^2). Let's define a basis (mathcal{B} { mathbf{b}_1 begin{bmatrix} 1 0 end{bmatrix}, mathbf{b}_2 begin{bmatrix} 1 1 end{bmatrix} }), and let (L: R^2 rightarrow R^2) be a linear transformation defined by (Lmathbf{b}_1 mathbf{b}_1) and (Lmathbf{b}_2 2mathbf{b}_2). The matrix representation of (L) in this basis is given by:

[ [L]_mathcal{B} begin{bmatrix} 1 0 0 2 end{bmatrix}. ]

Any matrix of (L) with respect to any basis is diagonalizable, with the result being the same. However, if the eigenspaces are not orthogonal, the matrix cannot be unitarily diagonalized. In this case, the eigenspaces (operatorname{span}{mathbf{b}_1}) and (operatorname{span}{mathbf{b}_2}) are not orthogonal, as (mathbf{b}_1 cdot mathbf{b}_2 eq 0).

Conclusion

In conclusion, while a matrix can be diagonalized in terms of eigenvalues and eigenvectors, the specific form of the diagonalization (the matrices (P) and (D)) can vary depending on the choice of eigenvectors and their arrangement. Therefore, matrix diagonalization is not unique. Furthermore, matrix diagonalization is not always unitary, and the matrix must have orthonormal eigenvectors to be unitarily diagonalizable.