Let 𝐰1, …, 𝐰n be an arbitrary basis of the subspace W ⊂ℝ^m. Let A=(𝐰1, …, 𝐰n) be the m ×n matrix

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Let 𝐰1, …, 𝐰n be an arbitrary basis of the subspace W ⊂ℝ^m....

Let $\mathbf{w}_1, \ldots, \mathbf{w}_n$ be an arbitrary basis of the subspace $W \subset \mathbb{R}^m$. Let $A=\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$ be the $m \times n$ matrix whose columns are the basis vectors, so that $W=\operatorname{img} A$ and $\operatorname{rank} A=n$. (a) Prove that the corresponding projection matrix $P=A\left(A^T A\right)^{-1} A^T$ is idempotent: $P^2=P$. (b) Prove that $P$ is symmetric. (c) Prove that $\operatorname{img} P=W$. (d) (e) Prove that the orthogonal projection of $\mathbf{v} \in \mathbb{R}^n$ onto $\mathbf{w} \in W$ is obtained by multiplying by the projection matrix: $\mathbf{w}=P \mathbf{v}$. (f) Show that if $A$ is nonsingular, then $P=\mathrm{I}$. How do you interpret this in light of part $(e)$ ? $(g)$ Explain why Exercise 4.4 .9 is a special case of this result. (h) Show that if $A=Q R$ is the factorization of $A$ given in Exercise 4.3 .32 , then $P=Q Q^T$. Why is $P \neq \mathrm{I}$ ?

Key Concepts

QR Factorization

QR factorization is a specific type of matrix factorization where a matrix is decomposed into the product of an orthogonal (or unitary) matrix Q and an upper triangular matrix R. In the context of projections, representing the projection matrix in terms of Q (i.e., P = QQ^T) leverages the orthonormal properties of Q to simplify the calculation and interpretation, though it may not yield the identity matrix unless the original matrix is square and invertible.

Matrix Factorization

Matrix factorizations, such as the QR factorization, decompose a matrix into simpler, often triangular or orthogonal components. Such decompositions help reveal structural properties of the matrix and can simplify computations, such as those involving projection matrices, by expressing the projection in terms of an orthogonal basis.

Orthogonal Projection

Orthogonal projection refers to the process of projecting a vector onto a subspace in such a way that the error (the difference between the original vector and its projection) is orthogonal to the subspace. This is accomplished using a projection matrix that minimizes the distance between the vector and the subspace.

Projection Matrix

A projection matrix is a linear operator that maps every vector in a space onto a subspace. It has the property that any vector that is already in the subspace remains unchanged when the projection is applied. It is typically constructed in a way that reflects the geometry of the subspace onto which the projection occurs.

Symmetry

A symmetric matrix is one that is equal to its transpose. In the context of projection matrices, symmetry guarantees that the projection preserves inner products and angles, which is essential for geometric interpretations such as orthogonal projections.

Idempotence

A matrix is idempotent if, when applied twice, it yields the same result as when applied once (i.e., P^2 = P). This property is fundamental to projection matrices because once a vector is projected onto the subspace, further applications of the projection do not alter it.

Column Space

The column space of a matrix is the subspace spanned by its column vectors. When a projection matrix is constructed from a basis of a subspace, its column space coincides with that subspace, ensuring that the projection maps vectors to this specific subset of the ambient space.

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