(a) Prove that if Q is an orthogonal matrix, then Q 𝐱=𝐱 for every vector 𝐱 ∈ℝ^n, where · denotes

### Part (a): Proving that if u005C( Q u005C) is an orthogonal matrix, then u005C( u005C|Q u005Cmathbf{x}u005C| u003D u005C|u005Cmat

(a) Prove that if Q is an orthogonal matrix, then Q 𝐱=𝐱 for...

(a) Prove that if $Q$ is an orthogonal matrix, then $\|Q \mathbf{x}\|=\|\mathbf{x}\|$ for every vector $\mathbf{x} \in \mathbb{R}^n$, where $\|\cdot\|$ denotes the standard Euclidean norm. (b) Prove the converse: if $\|Q \mathbf{x}\|=\|\mathbf{x}\|$ for all $\mathrm{x} \in \mathbb{R}^n$, then $Q$ is an orthogonal matrix.

Key Concepts

Orthogonal Matrix

An orthogonal matrix is a square matrix that satisfies the condition Q^T Q = I, where Q^T is the transpose of Q and I is the identity matrix. This condition ensures that the matrix preserves the inner product, thereby preserving the angles and lengths between vectors. Orthogonal matrices represent essential geometric transformations such as rotations and reflections in Euclidean space.

Euclidean Norm

The Euclidean norm of a vector in ?^n is defined as the square root of the sum of the squares of its components. It quantifies the length or magnitude of the vector and is a fundamental concept in geometry and linear algebra. The Euclidean norm is used to measure distances and to study the behavior of vectors under various linear transformations.

Norm Preservation under Linear Transformations

A linear transformation preserves the Euclidean norm if, for every vector x in ?^n, the norm of the transformed vector Qx is equal to the norm of x. This property is intimately linked to the concept of orthogonal matrices, as it characterizes the transformation as isometric (distance-preserving). Proving such norm preservation is crucial to both establishing and characterizing orthogonality of matrices.

Converse Proofs in Linear Algebra

In linear algebra, proving the converse of a statement is an important technique to establish equivalence between different definitions or properties. In this context, while it is straightforward to show that an orthogonal matrix preserves the norm, proving the converse—that any matrix preserving the norm must be orthogonal—requires demonstrating that norm preservation implies the matrix satisfies the defining condition of orthogonality (i.e., Q^T Q = I).

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