Question

If \(A\) is \(2 \times 2\) and diagonalizable, show that \(C(A)=\\{X \mid X A=A X\\}\) has dimension 2 or 4. [Hint: If \(P^{-1} A P=D,\) show that \(X\) is in \(C(A)\) if and only if \(P^{-1} X P\) is in \(C(D) .\)

Short answer

The dimension of \(C(A)\) is 2 or 4, depending on the eigenvalues of \(A\).

Step-by-step solution

Recognize the Set Definitions

The set \(C(A)\) represents all matrices \(X\) that commute with \(A\), meaning \(XA = AX\). This is called the centralizer of \(A\) in the space of \(2 \times 2\) matrices.

Use Diagonalizability

Given that \(A\) is diagonalizable, we can write \(A = PDP^{-1}\), where \(D\) is a diagonal matrix and \(P\) is the matrix of eigenvectors.

Transformations with Similar Matrices

If \(P^{-1}AP = D\), then any matrix \(X\) is in \(C(A)\) if and only if \(P^{-1}XP\) is in \(C(D)\), meaning \(P^{-1}XP\) must commute with \(D\).

Analyze Commutation with Diagonal Matrix

For a diagonal matrix \(D\), \(P^{-1}XP\) commutes with \(D\) if \((P^{-1}XP)D = D(P^{-1}XP)\). For \(2 \times 2\) matrices, this means the off-diagonal elements of \(P^{-1}XP\) are zero or \(D\) must have equal diagonal elements.

Determine Dimensions of \(C(D)\)

If \(D\) has distinct eigenvalues, the only matrices that commute with \(D\) are diagonal matrices themselves, giving dimension 2. If \(D\) has equal eigenvalues, any \(2 \times 2\) matrix commutes with \(D\), giving dimension 4.

Conclusion for \(C(A)\)

Since \(X\) commutes with \(A\) if and only if \(P^{-1}XP\) commutes with \(D\), \(C(A)\) has the same dimension as \(C(D)\). Thus, \(C(A)\) can have dimension 2 or 4.

Key concepts

Diagonalizable Matrices

A matrix is called diagonalizable if it can be expressed in the form \(A = PDP^{-1}\), where \(P\) is an invertible matrix, and \(D\) is a diagonal matrix. This transformation is very useful because diagonal matrices are easier to work with. The ease stems from the fact that when a matrix is diagonal, the calculations involving powers or exponential functions of matrices simplify substantially.

The matrices \(P\) and \(D\) have special roles: \(P\) contains eigenvectors of the matrix \(A\) as its columns, and \(D\) houses the corresponding eigenvalues along its diagonal. The eigenvectors are crucial because they provide the "directions" in which the matrix transformation scales, and the eigenvalues tell by how much it scales in these directions. The process of diagonalization helps us understand the essential characteristics of a matrix more clearly.

Diagonalizability also implies that the matrix's behavior can be analyzed through its eigenvalues and eigenvectors without resorting to complex calculations involving non-diagonal matrices. This property is especially beneficial when dealing with large matrices in applications across physics, computer science, and engineering, where efficient and simplified computations are desired.

Matrix Commutation

Commutation for matrices is a concept that stems from the idea of matrix multiplication order. For two matrices \(A\) and \(B\), they commute if \(AB = BA\). This property is not always true for matrices, unlike scalar values where multiplication is commutative by nature.

When matrices commute, it implies that one transformation followed by another yields the same result as reversing the order of these transformations. This feature is particularly noteworthy in linear algebra because it signals a certain compatibility between the transformations. For diagonalizable matrices, this concept ties into their structure: when a matrix can be diagonalized, it's often simpler to find which matrices will commute with either the original or its diagonal form.

• Commuting matrices typically share a relationship involving their eigenvectors.
• When a matrix \(A\) and another matrix \(X\) commute, \(X\) is often seen as a functional transformation (like scaling, rotating) that preserves the structure of \(A\).

In the context of the centralizer of matrices, it's about identifying all matrices that commute with a given matrix \(A\). This is crucial for understanding symmetries and invariances in systems represented by matrices.

Eigenvalues

An eigenvalue of a matrix is a scalar that describes the factor by which the corresponding eigenvector is scaled during a transformation. For a matrix \(A\), an eigenvalue \(\lambda\) satisfies the equation \(A\mathbf{v} = \lambda \mathbf{v}\), where \(\mathbf{v}\) is the eigenvector associated with \(\lambda\).

Eigenvalues are a foundational concept in linear algebra because they offer deep insights into the nature of the matrix's transformations. If a matrix is diagonalizable, its eigenvalues appear as the entries of its diagonal matrix form \(D\). This succinctly communicates how the matrix stretches or shrinks along the lines formed by its eigenvectors.

• Distinct eigenvalues often mean a simple structure for the associated matrix transformations. Specifically, diagonalization is straightforward with unique eigenvalues.
• If the eigenvalues are repeated or equal, it influences the types of matrices that will commute with the given matrix.

Thus, identifying eigenvalues ensures a clear understanding of the transformations facilitated by the matrix. They play a vital role in stability analysis, systems dynamics, and various fields such as quantum mechanics, where they describe fundamental characteristics of operators.