1 Basic concepts

Diagonalization is a technique in linear algebra that rewrites a square matrix in a form that is easier to analyze. The key idea is to choose a basis in which the linear transformation acts independently on each basis vector. In that basis, the matrix has nonzero entries only on the main diagonal, so computations such as repeated multiplication or exponentiation become simpler.

1.1 Square matrices

A square matrix has the same number of rows and columns. Such matrices represent linear transformations from a vector space to itself. This setting is essential for diagonalization because eigenvalues and eigenvectors are defined for transformations that map a space back into the same space.

1.2 Eigenvalues and eigenvectors

An eigenvector of a matrix is a nonzero vector whose direction is preserved under the transformation, up to scaling. The corresponding scale factor is called the eigenvalue. If a matrix has enough eigenvectors to form a basis, then the action of the matrix can be described very simply in that basis.

1.3 Change of basis

A change of basis replaces one coordinate system with another. For diagonalization, the new basis is chosen from eigenvectors. In the new coordinates, the matrix representation of the transformation may become diagonal, reflecting the independent scaling of each basis direction.

1.4 Similar matrices

Two matrices are similar when they represent the same linear transformation in different bases. Similar matrices have the same eigenvalues, trace, and determinant. Diagonalization is therefore a special case of similarity, where a matrix is similar to a diagonal matrix.

2 Diagonalizable matrices

A matrix is diagonalizable when it can be transformed into a diagonal matrix by a suitable change of basis. This property is not universal, since some matrices do not have enough linearly independent eigenvectors. When diagonalization is possible, it gives a compact description of the original matrix.

2.1 Definition of diagonalizability

A square matrix is diagonalizable if there exists an invertible matrix whose columns are eigenvectors of the matrix, such that conjugating by this matrix produces a diagonal matrix. Equivalently, the underlying vector space has a basis consisting entirely of eigenvectors. This basis is often called an eigenbasis.

2.2 Diagonal matrices

A diagonal matrix has all off-diagonal entries equal to zero. Its action on a vector is simple: each coordinate is multiplied by the corresponding diagonal entry. Because of this form, many matrix operations are straightforward, including taking powers and computing determinants.

2.3 Conditions for diagonalization

Whether a matrix is diagonalizable depends on the relationship between its eigenvalues and eigenvectors. Having eigenvalues alone is not sufficient; there must be enough independent eigenvectors to span the entire space. Several criteria help determine this.

2.3.1 Distinct eigenvalues

A matrix with n distinct eigenvalues in an n-dimensional space is diagonalizable. In this case, the eigenvectors associated with different eigenvalues are automatically linearly independent. This is one of the most useful practical tests for diagonalizability.

2.3.2 Eigenspace dimension

For each eigenvalue, the dimension of its eigenspace measures how many independent eigenvectors it contributes. A matrix is diagonalizable when the total number of independent eigenvectors equals the dimension of the space. If some eigenspaces are too small, diagonalization fails.

2.3.3 Minimal polynomial criteria

The minimal polynomial of a matrix encodes the simplest polynomial relation satisfied by the matrix. A matrix is diagonalizable if and only if its minimal polynomial splits into distinct linear factors, with no repeated roots. This criterion is often useful in more abstract settings.

3 Methods of diagonalization

Diagonalization is usually carried out by an explicit sequence of calculations. One first finds the eigenvalues, then computes eigenvectors, and finally assembles them into a change-of-basis matrix. The result can then be checked by direct multiplication.

3.1 Finding eigenvalues

Eigenvalues are obtained by solving the characteristic equation. This equation arises from the determinant of the matrix minus a scalar multiple of the identity matrix. Its roots, counted with algebraic multiplicity, are the eigenvalues of the matrix.

3.2 Finding eigenvectors

Once the eigenvalues are known, each one is substituted into the defining equation for eigenvectors. This produces a homogeneous linear system. The nonzero solutions form the eigenspace associated with that eigenvalue.

3.3 Constructing the diagonalizing matrix

If enough eigenvectors are found, they are arranged as columns of an invertible matrix. This matrix changes coordinates from the standard basis to the eigenbasis. Conjugating the original matrix by this change-of-basis matrix yields the diagonal matrix of eigenvalues.

3.4 Verifying the decomposition

After constructing the diagonalizing matrix, one checks that the decomposition is correct. This is done by multiplying the matrices in the appropriate order and confirming that the result is diagonal. Verification is especially important in computations involving repeated or nearly repeated eigenvalues.

4 Theorems and results

Several important theorems describe when diagonalization is possible and how it behaves under different mathematical settings. These results connect diagonalization to deeper structure in linear algebra and provide useful shortcuts for classification.

4.1 Spectral theorem

The spectral theorem states that certain classes of matrices, especially real symmetric or complex Hermitian matrices, can be diagonalized in a particularly well-behaved way. In these cases, the diagonalizing matrix can often be chosen to have orthonormal columns. This result is one of the most important in matrix theory.

4.2 Diagonalization over different fields

Diagonalizability depends on the field over which the matrix is considered. A matrix may be diagonalizable over the complex numbers but not over the reals if its eigenvalues are not all real. Thus, the choice of number system can change the outcome.

4.3 Invariant subspaces

An invariant subspace is a subspace that is mapped into itself by a linear transformation. Eigenvectors span one-dimensional invariant subspaces, and diagonalization decomposes the whole space into a direct sum of such pieces. This viewpoint helps explain why diagonal matrices are so simple.

4.4 Algebraic and geometric multiplicity

The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. The geometric multiplicity is the dimension of its eigenspace. For diagonalization, the geometric multiplicities must be large enough that their sum equals the size of the matrix.

5 Applications

Diagonalization is valuable because it turns difficult matrix operations into manageable ones. Many problems in algebra, differential equations, and applied mathematics become simpler when expressed in an eigenbasis.

5.1 Computing powers of matrices

If a matrix is diagonalizable, its powers can be computed by raising the diagonal entries to the desired power. This avoids repeated multiplication of the original matrix. The method is especially useful for large exponents.

5.2 Solving linear recurrences

Linear recurrence relations can often be rewritten in matrix form. When the associated matrix is diagonalizable, the recurrence can be solved by analyzing its eigenvalues. This approach gives explicit formulas and reveals long-term behavior.

5.3 Differential equations

Systems of linear differential equations with constant coefficients often lead to matrices that can be diagonalized. In an eigenbasis, the system separates into independent scalar equations. This separation makes it easier to write solutions in closed form.

5.4 Quadratic forms

Diagonalization is also used to simplify quadratic forms. By changing variables to an eigenbasis, cross terms may disappear, leaving a sum of squared terms weighted by eigenvalues. This is useful in geometry, optimization, and classification problems.

Some matrices admit especially elegant diagonalization, while others require related methods. These cases show both the strengths and the limits of diagonalization as a general tool.

6.1 Symmetric matrices

Real symmetric matrices have especially favorable properties. Their eigenvalues are real, and their eigenvectors can be chosen to be orthogonal. This makes them among the easiest matrices to diagonalize and one of the most studied classes in linear algebra.

6.2 Orthogonal diagonalization

Orthogonal diagonalization is a form of diagonalization in which the change-of-basis matrix is orthogonal. Because orthogonal matrices preserve lengths and angles, this process is geometrically stable and especially useful in applications involving real symmetric matrices.

6.3 Jordan canonical form

When a matrix is not diagonalizable, it may still be brought into a nearly diagonal form known as the Jordan canonical form. This form groups eigenvalues into blocks and records the failure of diagonalization. It generalizes the diagonal case and is important in advanced linear algebra.

6.4 Simultaneous diagonalization

Sometimes several matrices can be diagonalized using the same basis. This is called simultaneous diagonalization. It is possible only under additional compatibility conditions, and it is often linked to commutation and shared invariant subspaces.