Question

Let Q₁, Q₂ ∈ Rⁿˣᵐ be matrices with orthonormal columns, and let Q ∈ Rᵐˣᵐ be such that Q₁ = Q₂Q. Prove that Q is an orthogonal matrix.

Hint: Consider the product Qᵗ₁Q₁ = (Q₂Q)ᵗ(Q₂Q)

Answer 1

To prove that Q is an orthogonal matrix, we need to show that its transpose is equal to its inverse. By manipulating the given equation Q₁ = Q₂Q and using properties of transpose, we can show that QᵗQ = I, which proves that Q is an orthogonal matrix.

Step-by-step explanation:

To prove that Q is an orthogonal matrix, we need to show that its transpose is equal to its inverse.

Let's start by calculating Qᵗ₁Q₁. As given in the hint, Q₁ = Q₂Q, so Qᵗ₁Q₁ = (Q₂Q)ᵗ(Q₂Q). By distributing the transpose, we get Qᵗ(Q₂ᵗQ₂)Q = QᵗIQ = QᵗQ.

Since Q₁ has orthonormal columns, Qᵗ₁Q₁ = I.

Therefore, QᵗQ = I, which shows that Q is an orthogonal matrix.