Question

If \(P^{-1} A P\) and \(P^{-1} B P\) are both diagonal, show that \(A B=B A .\) [Hint: Diagonal matrices commute.]

Short answer

Since \(P^{-1} A P\) and \(P^{-1} B P\) are diagonal, \(A B = B A\) due to commutative property.

Step-by-step solution

Define Required Matrices and Their Properties

Let \(D_A = P^{-1} A P\) and \(D_B = P^{-1} B P\). By definition, both \(D_A\) and \(D_B\) are diagonal matrices.

Use the Commutative Property of Diagonal Matrices

Since both \(D_A\) and \(D_B\) are diagonal, they commute: \(D_A D_B = D_B D_A\).

Express Product Using Similarity Transformation

Given \(D_A D_B = P^{-1} A P P^{-1} B P\), simplify using the identity matrix \(P P^{-1} = I\) to get \(D_A D_B = P^{-1} A B P\). Similarly, \(D_B D_A = P^{-1} B A P\).

Establish Equivalence Using Similarity

Since \(D_A D_B = D_B D_A\), it follows from similar matrices that \(P^{-1} A B P = P^{-1} B A P\).

Conclude Equality of AB and BA

By applying cancellation, we deduce that \(A B = B A\). The matrix \(P\) and its inverse do not affect the equality of the products of \(A\) and \(B\).

Key concepts

Diagonal Matrices

Diagonal matrices are a special type of square matrices in which all the elements outside the main diagonal are zero. This means that only the elements from the top left to the bottom right (the diagonal) can contain non-zero values. Mathematically, for a matrix to be diagonal, its elements should satisfy the condition that if an element is not on the diagonal, it must be zero.

• If a matrix is denoted as \( D \), then \( D = [d_{ij}] \) is diagonal if \( d_{ij} = 0 \) for all \( i eq j \).
• They are both simple to manipulate and computationally efficient.
• The identity matrix is a common example of a diagonal matrix with all diagonal elements equal to one.

These matrices are essential in matrix theory due to their unique properties, such as ease of inversion when non-zero elements remain on the diagonal, and their ability to simplify complex operations.

Commutative Property

The commutative property is a fundamental aspect of mathematics that states the order in which two elements are multiplied does not affect the outcome. For diagonal matrices, this property shines brightly.

• If \( D_A \) and \( D_B \) are diagonal matrices, then \( D_A D_B = D_B D_A \).
• This behavior simplifies operations significantly since the rows and columns that interact have corresponding zero elements, reducing abrupt calculations.
• Commutative property in diagonal matrices assures that reordering matrices in multiplication will not change the result, which is not usually valid for non-diagonal matrices.

The fact that diagonal matrices commute allows for significant simplifications, especially in proving equalities such as \( AB = BA \), confirming that these matrices hold symmetrical characteristics in mathematical operations.

Similarity Transformation

Similarity transformation is a process where one matrix is transformed into another matrix of the same dimensions via a particular form of equivalence. If two matrices \( A \) and \( B \) are similar under a matrix \( P \), then there exists an invertible matrix \( P \) such that:

• \( D_A = P^{-1} A P \)
• \( D_B = P^{-1} B P \)

In the context of diagonal matrices, similarity transformations are profoundly useful in simplifying complex matrices to reveal core properties. They are particularly useful because they maintain the characteristics of the original matrix, such as eigenvalues, while packaging them into a more accessible form (like a diagonal matrix).
One of the significant aspects of this transformation is that it helps to establish equality between matrix products post-transformation. Hence, when \( D_A D_B = D_B D_A \) is deduced to establish \( AB = BA \) using similarity transformations, it highlights the underlying commutative property preserved through the transformation. This shows that similarity transformations are a powerful tool in exploring and uncovering matrix relationships.