Question

a. If \(J\) is the \(4 \times 4\) matrix with every entry 1 , show that \(I-\frac{1}{2} J\) is self-inverse and symmetric. b. If \(X\) is \(n \times m\) and satisfies \(X^{T} X=I_{m},\) show that \(I_{n}-2 X X^{T}\) is self-inverse and symmetric.

Short answer

Both matrices are self-inverse and symmetric as shown.

Step-by-step solution

Understanding the Problem

We are tasked with proving that certain matrices, specifically \(I - \frac{1}{2} J\) and \(I_{n} - 2 X X^{T}\), where \(J\) is a 4x4 matrix with all entries as 1, and \(X\) is an \(n \times m\) matrix such that \(X^T X = I_m\), are self-inverse and symmetric.

Definitions and Simplifications for Part a

First, recognize that \(I\) is the identity matrix and given \(J\) is a 4x4 matrix with all elements as 1, \(J\) can be represented as \(J = \begin{bmatrix} 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \end{bmatrix}\). We need to investigate the properties of \(I - \frac{1}{2} J\).

Showing Symmetry for Part a

A matrix is symmetric if it equals its transpose. By examining \(I - \frac{1}{2} J\), we note this is already symmetric because both \(I\) and \(J\) are symmetric. Thus, \( (I - \frac{1}{2} J)^T = I - \frac{1}{2} J\).

Showing Self-Inverse Property for Part a

To prove self-inverse, we need \((I - \frac{1}{2} J)^2 = I\). Compute: \((I - \frac{1}{2} J)(I - \frac{1}{2} J) = I - (I, \frac{1}{2} J + \frac{1}{2} J) + \frac{1}{4} J^2\). Note that \(J^2 = 4J\) because every row and column sum of \(J\) is 4. Simplifying:\[ I - J + \frac{1}{4} \times 4J = I - J + J = I \]. Thus: \((I - \frac{1}{2} J)^2 = I\).

Definitions and Simplifications for Part b

Now, consider matrix \(X\) such that \(X^T X = I_m\) and focus on \(I_n - 2 X X^T\). We need to demonstrate its symmetry and self-inverseness.

Showing Symmetry for Part b

A matrix is symmetric if it equals its transpose, meaning \((X X^T)^T = X X^T\) because \((X X^T)^T = X^T X^T = X X^T\). This implies \(I_n - 2X X^T\) is symmetric since \((I_n - 2X X^T)^T = I_n - 2X X^T\).

Showing Self-Inverse Property for Part b

To prove self-inverse, demonstrate \((I_n - 2X X^T)^2 = I_n\). Calculate: \((I_n - 2X X^T)(I_n - 2X X^T) = I_n - 2 (I_nX X^T + X X^T I_n) + 4X X^T X X^T\). Knowing \(X^T X = I_m\) implies \(X X^T X X^T = X X^T\), resulting in\[ I_n - 4X X^T + 4X X^T = I_n \]. Thus, it satisfies \((I_n - 2X X^T)^2 = I_n\).

Conclusion: Analyzing Results

We have shown for both parts that the matrices \(I - \frac{1}{2} J\) and \(I_n - 2 X X^T\) are symmetric (since they equal their respective transposes) and self-inverse (since their square equals the identity matrix).

Key concepts

Symmetric Matrices

A symmetric matrix is quite straightforward. If you swap its rows and columns, it looks exactly the same. This means that a matrix is symmetric if it equals its transpose. Think of it as a mirror image where the matrix reflects perfectly over its diagonal.
Here are some important points about symmetric matrices:

• A matrix is symmetric if \(A = A^T\).
• Symmetric matrices are always square, meaning they have the same number of rows and columns.
• The diagonal elements of a symmetric matrix can be any number, while the non-diagonal elements must mirror each other.

Understanding symmetric matrices is important because they're everywhere in science and engineering. If you're dealing with anything involving vectors and transformations, you'll likely see them often.

Self-Inverse Matrices

Self-inverse, or involutory matrices, have a unique characteristic. Multiplying the matrix by itself gives you the identity matrix.
Here's how it works:

• If a matrix \(A\) is self-inverse, it follows that \(A^2 = I\), where \(I\) is the identity matrix.
• This special property means that performing the transformation twice returns you to the original state.
• Such matrices are beneficial in computations where reversing an operation is needed.

Self-inverse matrices split into two camps—those involving rotations and reflections in geometric spaces. These matrices are like magic undo buttons that bring transformations back to their starting point.

Identity Matrix

Imagine the identity matrix as the number 1 for matrices. When you multiply any matrix by the identity matrix, it remains unchanged. This type of matrix is very special in matrix theory.
Here are its features

• An identity matrix is always square, so it has the same number of rows and columns.
• It comprises all ones on the main diagonal and zeros elsewhere.
• Symbolically, for a matrix of size \(n \times n\), the identity matrix is represented as \(I_n\).

The identity matrix is crucial when solving equations or performing operations because it acts neutrally—it preserves the structure of other matrices during multiplication.

Transpose of a Matrix

Transposing a matrix is like flipping a pancake: what was once horizontal becomes vertical and vice versa. This operation is simple yet essential in matrix theory.
Key points to transposing a matrix:

• The transpose of a matrix \(A\), denoted \(A^T\), is obtained by exchanging rows with columns.
• If \(A\) is an \(m \times n\) matrix, then \(A^T\) becomes an \(n \times m\) matrix.
• For square matrices, the transpose is square too.

The transpose is instrumental in changing perspectives within problems, particularly when working with linear transformations, making life in matrix land much more dynamic.