Proving the Sum of Two Orthogonal Matrices is Not Orthogonal

" "In this article, we will explore the rigorous mathematical proof that the sum of two orthogonal matrices is not necessarily an orthogonal matrix. We will start with the definition of an orthogonal matrix and proceed to analyze the conditions under which the sum of two such matrices can or cannot be orthogonal.

Definition of an Orthogonal Matrix

An orthogonal matrix is a square matrix A that satisfies the condition:

AT A I

where AT is the transpose of A, and I is the identity matrix. This condition can be equivalently written as:

AT A-1

Introduction to the Problem

To prove that the sum of two orthogonal matrices is not necessarily orthogonal, we will first assume two orthogonal matrices A and B. By definition, we have:

AT A I BT B I

Let C A B be the sum of these two matrices. For C to be orthogonal, it must satisfy:

CT C I

Mathematical Derivation

Let's calculate CT C:

CT (A B)T AT BT

Then,

CT C (AT BT) (A B)

Expanding this expression, we get:

CT C ATA ATB BTA BTB

Substituting the orthogonality conditions:

CT C I ATB BTA I

Simplifying further:

CT C 2I ATB BTA

For C to be orthogonal, we need:

2I ATB BTA I

Reorganizing this equation:

ATB BTA -I

Conclusion

The condition ATB BTA -I is not generally satisfied for arbitrary orthogonal matrices A and B. Therefore, the sum of two orthogonal matrices is not necessarily an orthogonal matrix. The only case where the sum is orthogonal is when one of the matrices is the negative of the other, i.e., B -A.

Counterexample

To further illustrate this point, let's consider specific matrices:

Matrix A and Matrix B

Let A and B be two orthogonal matrices:

A begin{bmatrix} 0.8 0.6 0.6 -0.8 end{bmatrix} B begin{bmatrix} 0.96 0.28 -0.28 0.96 end{bmatrix}

Then, the sum C A B is:

C begin{bmatrix} 1.76 0.88 0.32 0.16 end{bmatrix}

Checking if C is orthogonal:

CTC begin{bmatrix} 1.76 0.32 0.88 0.16 end{bmatrix}begin{bmatrix} 1.76 0.88 0.32 0.16 end{bmatrix} begin{bmatrix} 3.2672 1.7632 1.7632 0.9104 end{bmatrix}

Since CTC ≠ I, C is not orthogonal.

This counterexample demonstrates that the sum of two orthogonal matrices is not guaranteed to be orthogonal.

Summary

Given two orthogonal matrices A and B, the sum A B is not necessarily an orthogonal matrix. The only case where the sum is orthogonal is when B -A. In all other cases, the sum does not satisfy the orthogonality condition.