Question

The trace of a matrix is the sum of the elements on the main diagonal. Show that the trace is not changed by cyclic permutation of the matrices, that is, \(\operatorname{Tr}(A B C)=\operatorname{Tr}(C A B)=\) Tr \((B C A)\). [Caution: \(\operatorname{Tr}(A B C) \neq \operatorname{Tr}(A C B)\) in general. ]

Short answer

The trace of the matrix product remains unchanged by cyclic permutation: \(\text{Tr}(ABC) = \text{Tr}(CAB) = \text{Tr}(BCA)\).

Step-by-step solution

Understand the Definition of Trace

The trace of a matrix is the sum of the elements on its main diagonal. For a matrix A, it is denoted by \(\text{Tr}(A)\) and is calculated as: \(\text{Tr}(A) = \begin{pmatrix} a_{11} & a_{12} & \text{...} \ a_{21} & a_{22} & \text{...} \ \text{...} & \text{...} & \text{...} \end{pmatrix} = a_{11} + a_{22} + \text{...}\).

Use the Property of Trace with Multiplication

The trace of product matrices has the cyclic permutation property which states \(\text{Tr}(ABC) = \text{Tr}(CAB) = \text{Tr}(BCA)\). We'll demonstrate it for three matrices A, B, and C.

Consider the Element Definition

For matrices A, B, and C where A is \(n \times n\), B is \(n \times n\), and C is \(n \times n\), the \(i,j\)-th element of the product \(ABC\) is given by \sum_{k=1}^{n} a_{ik} b_{kj} c_{ji}\.

Calculate the Trace of ABC

The trace of \(ABC\) is calculated as \(\text{Tr}(ABC) = \sum_{i=1}^{n} \sum_{k=1}^{n} \sum_{j=1}^{n} a_{ik} b_{kj} c_{ji}\). This is the sum of the diagonal elements of the resulting matrix of \(ABC\).

Apply Cyclic Permutation

To show \(\text{Tr}(CAB) = \text{Tr}(BCA)\), notice that \(\text{Tr}(CAB) = \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} c_{ii} a_{ij} b_{jk}\) involves a similar rearrangement of elements which sums the same diagonal elements.

Conclude the Cyclic Permutation

Thus, cyclic permutations of the matrices do not alter the set of products whose sum forms the trace. Hence, \(\text{Tr}(ABC) = \text{Tr}(CAB) = \text{Tr}(BCA)\).

Key concepts

cyclic permutation

To understand cyclic permutation in matrix operations, it's essential to know what a permutation is. A permutation refers to rearranging elements in a particular order. In the context of matrices, cyclic permutation means rotating the order of matrix products in a cycle.

For example, if you have matrices A, B, and C, a cyclic permutation involves the following:

• From ABC to CAB
• From CAB to BCA
• From BCA back to ABC

This order rotation doesn't affect the calculations of the trace. However, not all permutations are cyclic. For instance, the permutation from ABC to ACB is not cyclic, and the trace could differ.

Why is this important? Because understanding cyclic permutation helps simplify complex matrix operations and proofs, especially those involving traces of matrices. It ensures that while the positions of matrices change, their overall cyclic properties and results remain consistent.

trace of matrix

The trace of a matrix is a fundamental concept in linear algebra. It is defined as the sum of all elements on the main diagonal of a square matrix.

Consider a matrix A:
\( A = \begin{pmatrix} a_{11} & a_{12} & \dots \ a_{21} & a_{22} & \dots \ \dots & \dots & \dots \end{pmatrix} \).
The trace of matrix A, denoted \( \operatorname{Tr}(A) \), is:
\( \operatorname{Tr}(A) = a_{11} + a_{22} + \dots \).

The trace has important properties which simplify many matrix computations. One key property is its invariance under cyclic permutation – meaning the trace remains the same if the order of matrix multiplication is cycled, such as:

• \( \operatorname{Tr}(ABC) = \operatorname{Tr}(CAB) = \operatorname{Tr}(BCA) \)

However, this property does not hold for non-cyclic permutations (e.g., \( \operatorname{Tr}(ABC) eq \operatorname{Tr}(ACB) \)). Understanding and applying the trace of matrix helps analyze and simplify various matrix equations and identities.

matrix multiplication

Matrix multiplication is one of the core operations in linear algebra. If you have two matrices A and B, their product, denoted AB, is a new matrix formed by combining rows of A with columns of B.

To multiply matrices A (of size m by n) and B (of size n by p), the resulting matrix AB will be of size m by p. The element in the i-th row and j-th column of AB is computed as:
\((AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}\).

Here's a simple breakdown:

• For each element in the resulting matrix, match the row from A with the corresponding column from B.
• Multiply each pair of elements and sum these products to get the final element for the resulting position.
  
  
  When dealing with the trace and cyclic permutations, matrix multiplication becomes a key step. By computing products and summing diagonals in a certain order (using the properties of multiplication and trace), you can prove the invariance of the trace under cyclic permutations.
  
  This understanding is essential for linear algebra, computer graphics, systems of linear equations, and more advanced topics in mathematics and engineering.