Question

Let \(A\) be \(n \times n\) with \(n\) distinct real eigenvalues. If \(A C=C A,\) show that \(C\) is diagonalizable.

Short answer

Matrix \(C\) is diagonalizable because it commutes with diagonalizable matrix \(A\) with distinct eigenvalues.

Step-by-step solution

Understanding the Problem

First, we need to understand the properties of matrix \(A\) and matrix \(C\). We know \(A\) is an \(n \times n\) matrix with \(n\) distinct eigenvalues, which makes \(A\) diagonalizable. We are given the commutation relation \(AC = CA\) and want to show that \(C\) is also diagonalizable.

Diagonalizing Matrix A

Because \(A\) has \(n\) distinct real eigenvalues, it is diagonalizable. Hence, there exists an invertible matrix \(P\) such that \(P^{-1}AP = D\), where \(D\) is a diagonal matrix with the eigenvalues of \(A\) on its diagonal.

Using Commutativity AC = CA

Since \(AC = CA\), the matrices \(A\) and \(C\) share the same eigenvectors. If \(v\) is an eigenvector of \(A\) corresponding to eigenvalue \(\lambda\), then \(Av = \lambda v\). Because \(AC = CA\), it follows that \(C(Av) = A(Cv)\), which implies \(Cv\) is also an eigenvector of \(A\) with the same eigenvalue \(\lambda\).

Constructing Matrix C in Eigenbasis of A

Write \(C\) in the basis of the eigenvectors of \(A\). Let \(V\) be the matrix formed by the eigenvectors of \(A\). We have \(A = VDV^{-1}\), and \(C = VGV^{-1}\), where \(G\) is computed such that \(V^{-1}AV = D \), and \(V^{-1}CV = G\).From \(AC = CA\), we can deduce \(DG = GD\).

Diagonalizing G

Since \(DG = GD\), \(D\) and \(G\) are simultaneously diagonalizable. \(D\) is already diagonal, the commutation implies that \(G\) has the same eigenvectors as \(D\), hence \(G\) is also diagonal. Thus, \(C = VGV^{-1}\) where \(G\) is diagonal, implying \(C\) is diagonalizable.

Key concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra relevant to many areas like physics and engineering.
An eigenvalue is a scalar that indicates how much an eigenvector is stretched during a linear transformation by a matrix. An eigenvector is non-zero and changes only in scale, not direction, when multiplied by this matrix.
Given a matrix \(A\), if \(v\) is an eigenvector, and \(\lambda\) (lambda) is its corresponding eigenvalue, then:

• Equation: \(Av = \lambda v\)
• \(v\) is non-zero
• \(\lambda\) shows how the eigenvector scales under the matrix \(A\).

Understanding these helps in simplifying complex matrix related problems, as we often work towards finding such vectors and values. They help identify the "axes" of transformation, making them crucial for tasks like diagonalization.

Matrix Commutation

Matrix commutation refers to the property where two matrices commute if their product is the same in either order.
For matrices \(A\) and \(C\), the rule is: \(AC = CA\). This is an essential property when discussing diagonalizability and simultaneous diagonalization. If two matrices commute, it implies they can be diagonalized in the same basis, sharing a common set of eigenvectors.

• Commutating matrices: Often share eigenvectors
• This property is used to simplify matrix computations
• Helps relate two or more linear transformations

This property is particularly useful in various applications, including physics, where it relates to the symmetrical properties of quantum systems.

Simultaneous Diagonalization

Simultaneous diagonalization is a technique that allows two commuting matrices to be diagonalized at the same time.
When matrices \(A\) and \(C\) commute (\(AC = CA\)), they share a common set of eigenvectors, allowing them to be simultaneously diagonalized. In practice, this means that there exists a single invertible matrix \(V\) such that both \(A\) and \(C\) can be diagonalized using \(V\).

• Matrix \(A\) and \(C\) commute: \(AC = CA\)
• Use same eigenvectors for diagonalization
• Diagonals reflect the system's intrinsic properties

Simultaneous diagonalization is powerful as it lets us analyze and solve complex systems more straightforwardly, turning complicated matrix equations into simple diagonal ones.