Question

We call a square matrix \(E\) a projection matrix if \(E^{2}=E=E^{T} .\) (See Exercise 8.1.17.) a. If \(E\) is a projection matrix, show that \(P=I-2 E\) is orthogonal and symmetric. b. If \(P\) is orthogonal and symmetric, show that \(E=\frac{1}{2}(I-P)\) is a projection matrix. c. If \(U\) is \(m \times n\) and \(U^{T} U=I\) (for example, a unit column in \(\mathbb{R}^{n}\) ), show that \(E=U U^{T}\) is a projection matrix.

Short answer

a. P is both orthogonal and symmetric. b. E is a projection matrix. c. E is a projection matrix.

Step-by-step solution

Show that P is Orthogonal

To prove that a matrix is orthogonal, we need to show that its transpose is its inverse, i.e., \(P^{T}P = I\). Given \(P = I - 2E\), First, find \(P^{T}\): \[ P^{T} = (I - 2E)^{T} = I^{T} - (2E)^{T} = I - 2E \]because both \(I\) and \(E\) are symmetric. Hence, \(P = P^{T}\). Now compute \(PP^{T}\): \[ PP^{T} = (I - 2E)(I - 2E) = I - 2E - 2E + 4E^{2} \] Since \(E^{2} = E\), \[ PP^{T} = I - 4E + 4E = I \].Thus, \(P\) is orthogonal.

Show that P is Symmetric

A matrix is symmetric if it is equal to its transpose. From the previous step, we established that \(P = P^{T}\). Thus, \(P\) is symmetric.

Show that E is a Projection Matrix from P

Start with the condition that \(P = I - 2E\) is orthogonal and symmetric.Express \(E\) in terms of \(P\):\[ E = \frac{1}{2}(I - P) \]Now show \(E^{2} = E\): \[ E^{2} = \left(\frac{1}{2}(I - P)\right)\left(\frac{1}{2}(I - P)\right) = \frac{1}{4}(I - 2P + P^{2}) \] Since \(P^{2} = I\) because \(P\) is orthogonal, it simplifies to:\[ E^{2} = \frac{1}{4}(I - 2P + I) = \frac{1}{4}(2I - 2P) = \frac{1}{2}(I - P) = E \].Thus, \(E^{2} = E\).Also confirm that \(E = E^{T}\): \[ E^{T} = \left(\frac{1}{2}(I - P)\right)^{T} = \frac{1}{2}(I^{T} - P^{T}) = \frac{1}{2}(I - P) = E \].Thus, \(E\) is symmetric.

Show E = UU^T is a Projection Matrix

Given a matrix \(U\) such that \(U^T U = I\), consider \(E = U U^T\).Verify \(E^{2} = E\):\[ E^{2} = (U U^T)(U U^T) = U (U^T U) U^T = U I U^T = U U^T = E \].Check if \(E\) is symmetric:\[ E^{T} = (U U^T)^{T} = (U^T)^{T} U^{T} = U U^T = E \].Thus, \(E^{2} = E\) and \(E = E^{T}\), making \(E\) a projection matrix.

Key concepts

Orthogonal Matrix

An orthogonal matrix is a special type of square matrix. In simple terms, a matrix is orthogonal if its transpose is equal to its inverse. This means that when you multiply the matrix by its transpose, you get the identity matrix. The identity matrix is a matrix with 1s on the diagonal and 0s elsewhere, acting similar to the number 1 in multiplication. An orthogonal matrix preserves the dot product, which means it keeps lengths and angles unchanged when transformed.
To check if a matrix is orthogonal, verify that \(P^TP = I\). If this holds true, the matrix is indeed orthogonal. Orthogonal matrices are particularly useful in tasks that involve rotation or reflection, as they maintain the structure of the data.

Symmetric Matrix

A symmetric matrix is one that is identical to its transpose. If you flip it over its main diagonal, it remains unchanged. This property is useful because it often simplifies matrix computations and analyses.
For a matrix \(A\) to be symmetric, \(A = A^{T}\). Symmetric matrices are common in various fields like physics and computer graphics because they represent systems or transformations that don't change when flipped or rotated along the main diagonal.
In the context of projection matrices and the task at hand, proving that a matrix is symmetric can often make verifying other properties, such as projection or orthogonality, straightforward.

Matrix Transpose

The transpose of a matrix is achieved by swapping its rows and columns. If you have a matrix \(A\) and you take its transpose, denoted \(A^T\), each element \(a_{ij}\) in the \(i\)-th row and \(j\)-th column of \(A\) becomes \(a_{ji}\) in \(A^T\).
This operation is particularly useful in many mathematical computations, including solving linear equations and optimizing functions. The transpose is also important when dealing with orthogonal matrices, as mentioned earlier, because for these matrices, the transpose equates to the inverse.
Understanding how to compute and utilize the transpose can simplify many problems in linear algebra, particularly those involving symmetric or projection matrices.

Matrix Inverse

The inverse of a matrix is like finding the reciprocal of a number. If you multiply a matrix by its inverse, the result is the identity matrix. Not all matrices have inverses, but when they do, it plays a crucial role in solving matrix equations.
For a matrix \(A\), its inverse is denoted as \(A^{-1}\), and the key property is \(AA^{-1} = I\) where \(I\) is the identity matrix. Calculating the inverse is useful in numerous applications including solving systems of linear equations, computer graphics, and more. However, computing the inverse can be computationally intensive and not always possible if the matrix is not square or has a determinant of zero.
Understanding and using the inverse is essential for dealing with orthogonal matrices, as the transpose of such a matrix is also its inverse.