(a) Show that if D=( a 0 0 b ) is a 2 ×2 diagonal matrix with a ≠b, then the o

step 1 For this exercise, we have a b sub one equal to a b c d times a two by matrix one zero zero zero

(a) Show that if D=( a 0 0 b ) is a 2 ×2...

(a) Show that if $D=\left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right)$ is a $2 \times 2$ diagonal matrix with $a \neq b$, then the only matrices that commute (under matrix multiplication) with $D$ are other $2 \times 2$ diagonal matrices. (b) What if $a=b$ ? (c) Find all matrices that commute with $D=\left(\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right)$, where $a, b, c$ are all different. (d) Answer the same question for the case when $a \neq b=c$. (e) Prove that a matrix $A$ commutes with an $n \times n$ diagonal matrix $D$ with all distinct diagonal entries if and only if $A$ is a diagonal matrix.

Key Concepts

Distinct Eigenvalues

When a diagonal matrix has distinct diagonal entries, each entry represents a unique eigenvalue, thereby ensuring that the corresponding eigenspaces are one-dimensional. This property forces any matrix commuting with the diagonal matrix to act independently on each eigenspace, typically restricting the commutant to be diagonal as well.

Centralizer

The centralizer (or commutant) of a matrix D is the set of all matrices that commute with D. In the context of diagonal matrices with distinct diagonal entries, the centralizer typically consists of only the diagonal matrices, since any off?diagonal entry in a commuting matrix would contradict the uniqueness of the eigenvalues associated with different diagonal positions.

Scalar Matrix

A scalar matrix is a special case of a diagonal matrix in which all the diagonal entries are equal. Since a scalar matrix is a multiple of the identity matrix, it commutes with every matrix. This fact illustrates one extreme case in the study of matrix commutativity, contrasting with diagonal matrices with distinct entries.

Diagonal Matrix

A diagonal matrix is a square matrix whose entries outside the main diagonal are all zero. Its simplicity facilitates many computations in linear algebra, especially when considering eigenvalues and eigenvectors, as the eigenvalues of a diagonal matrix are exactly the entries along its main diagonal.

Matrix Commutation

Matrix commutation refers to the property that two matrices A and B satisfy AB = BA. This concept is important in linear algebra because it often reveals structural similarities between the matrices, such as sharing invariant subspaces or being simultaneously diagonalizable under certain conditions.

Block Diagonal Structure

In cases where a diagonal matrix has some repeated diagonal entries (for example, when one entry is distinct and the rest are equal), the matrices that commute with it can exhibit a block diagonal structure. This means that while the overall matrix might not be completely diagonal, it can be partitioned into smaller blocks (or submatrices), each acting on the invariant subspace associated with the repeated eigenvalue.

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