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Chapter 3: Problem 18

If \(A=Q R\), find a simple formula for the projection matrix \(P\) onto the column space of \(A\).

Short Answer

Expert verified
The projection matrix is \( P = QQ^T \).

Step by step solution

01

Understanding the Problem

We are given a matrix equation \( A = QR \) where \( Q \) and \( R \) come from the QR decomposition of \( A \). This decomposition tells us that \( Q \) is an orthonormal matrix and \( R \) is an upper triangular matrix. Our goal is to find the projection matrix \( P \) that projects any vector onto the column space of \( A \).
02

Recall the Projection Matrix Form

The projection matrix \( P \) onto the column space of a matrix \( A \) is generally given by the formula \( P = A(A^TA)^{-1}A^T \). This formula ensures that any vector multiplied by \( P \) is projected onto the column space of \( A \).
03

Utilize QR Decomposition

Substitute \( A = QR \) into the formula for \( P \). We substitute to obtain \( A^T = (QR)^T = R^TQ^T \). This makes the projection formula become \( P = QR(R^TQ^TRR^T)^{-1}(R^TQ^T) \).
04

Simplify the Expression

Remember that \( Q^TQ = I \), where \( I \) is the identity matrix, because \( Q \) is orthonormal. This simplifies the inside of the inverse in our expression: \( (R^TR)^{-1} \). As a result, the projection matrix becomes \( P = Q(RR^T)^{-1}Q^T \).
05

Simplify Further

Since \( Q \) is orthonormal, \( QQ^T = I \). Therefore, the final form of the projection matrix simplifies to \( P = QQ^T \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Projection Matrix
A projection matrix is a very handy tool in linear algebra. Essentially, it "projects" or maps a vector onto a subspace, such as the column space of another matrix.
To understand this better:
  • Imagine a light source above a shape, casting a shadow onto a flat surface below. The shadow represents the projection of the shape onto that surface.
  • In mathematics, this is like taking a vector and projecting it onto the line or plane spanned by the columns of a matrix.
  • The formula for the projection matrix onto the column space of a matrix \( A \) is \( P = A(A^TA)^{-1}A^T \).
This formula ensures that when a vector is multiplied by \( P \), it lies within the column space of \( A \). In the case of QR decomposition, the process simplifies dramatically because of the orthonormal properties of \( Q \). This means the projection onto the column space of \( A \) can simply be represented as \( P = QQ^T \).
Orthonormal Matrix
An orthonormal matrix, like the \( Q \) in the QR decomposition, has unique properties that simplify calculations. These matrices have columns that are both orthogonal and of unit length.
Let's break this down:
  • An orthogonal set of vectors means they are at right angles to each other. In practical terms, this means the dot product between any two distinct columns of \( Q \) is zero.
  • Unit length means each vector (column in \( Q \)) has a magnitude of one 鈥� making the matrix not just orthogonal, but orthonormal.
  • An essential property is \( Q^TQ = I \), where \( I \) is the identity matrix. This shows how operations with orthonormal matrices can result in very neat simplifications.
This simplifies equation calculations significantly. For example, in the QR decomposition of a matrix \( A \), the orthonormal property of \( Q \) allows the projection matrix to be expressed as \( P = QQ^T \) with fewer steps.
Column Space
The column space of a matrix is a fundamental concept in understanding linear transformations and equations in vector spaces. It encompasses all linear combinations of the matrix's columns.
Here's how it works:
  • The column space is the vector span of all the columns in a matrix \( A \). Essentially, if you combine the columns in various ways 鈥� multiplied by scalars and summed 鈥� you get every possible vector in the column space.
  • It's important because it defines an essential part of the behavior of the matrix when used to transform other vectors.
  • Solutions to linear systems \( Ax = b \) exist only if \( b \) is in the column space of \( A \).
Using the QR decomposition, the column space of \( A \) is critical. This is because every projection of another vector onto this space relies on these column vectors. It directly affects the way the projection matrix \( P \) is formed as \( QQ^T \) in a simplified form due to the orthonormal property of \( Q \).

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Most popular questions from this chapter

Project the vector \(b=(1,1)\) onto the lines through \(a_{1}=(1,0)\) and \(a_{2}=(1,2)\). Draw the projections \(p_{1}\) and \(p_{2}\) and add \(p_{1}+p_{2}\). The projections do not add to \(b\). because the \(a\) 's are not orthogonal.

(a) If \(P=P^{\mathrm{T}} P\), show that \(P\) is a projection matrix. (b) What subspace does the matrix \(P=0\) project onto?

Suppose \(A\) is a symmetric matrix \(\left(A^{T}=A\right)\). (a) Why is its column space perpendicular to its nullspace? (b) If \(A x=0\) and \(A z=5 z\), which subspaces contain these "eigenvectors" \(x\) and \(z\) ? Symmetric matrices have perpendicular eigenvectors (see Section 5.5).

By setting the derivative to zero, find the value of \(b_{1}\) that minimizes $$ \left\|b_{1} \sin x-\cos x\right\|^{2}=\int_{0}^{2 \pi}\left(b_{1} \sin x-\cos x\right)^{2} d x $$ Compare with the Fourier coefficient \(h\)

From the nonorthogonal \(a, b, c\), find orthonormal vectors \(q_{1}, q_{2}, q_{3}\) : $$ a=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right], \quad c=\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right] $$
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