Question

A matrix that we obtain from the identity matrix by writing its rows in a different order is called a permutation matrix. Show that every permutation matrix is orthogonal.

Short answer

Permutation matrices are orthogonal because their transpose equals their inverse.

Step-by-step solution

Define a Permutation Matrix

A permutation matrix is obtained by rearranging the rows of an identity matrix. For example, rearranging the rows of a 3x3 identity matrix can give us a permutation matrix such as \[ P = \begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix} \].

Define Orthogonal Matrix

A matrix is orthogonal if its transpose is equal to its inverse, i.e., \( A^T = A^{-1} \). An orthogonal matrix \( A \) satisfies \( A^T A = I \). This means the dot product between different columns of the matrix is zero and between similar columns is one.

Verify Transpose and Inverse Property

For permutation matrix \( P \), we take the transpose \( P^T \) which is simply swapping rows with columns. Since a permutation matrix involves reordering the identity matrix, when we swap rows and columns, the structure remains such that different columns remain orthogonal, and each column has exactly one '1' and the rest '0's.

Calculate \( P^T P \)

To demonstrate orthogonality, calculate \( P^T P \): \[ P^T P = \begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = I \]. Repeating this operation gives us the identity matrix \( I \), confirming \( P^T = P^{-1} \).

Conclude Orthogonal Property

Having established that \( P^T P = I \), it confirms that permutation matrix \( P \) is orthogonal, as it satisfies the defining condition for orthogonal matrices: transpose equals inverse.

Key concepts

Orthogonal Matrix

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors. This means each column vector in the matrix is perpendicular to the others, and each has a magnitude of one. For a matrix to be orthogonal, when you multiply it by its transpose, you should get the identity matrix as the result.
This gives orthogonal matrices a special property: the transpose of the matrix is also its inverse. This is written in mathematical terms as:

• Given an orthogonal matrix \( A \), we have \( A^T A = I \).
• The inverse is \( A^{-1} = A^T \).

So, if you hear that a matrix is orthogonal, you know that it stays unchanged by these multi-operations with its transpose.

Matrix Transpose

The transpose of a matrix is a new matrix, achieved by swapping the rows and columns of the original matrix. If you have a matrix \( A \), its transpose is noted as \( A^T \).
Imagine you have a row matrix as part of your matrix \( A \). To transpose, write this row as a column in \( A^T \). Similarly, if you start with a column, it becomes a row in \( A^T \). For example:

• Original matrix: \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \)
• Transpose: \( A^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix} \)

This swapping technique modifies the original data structure but keeps the information intact. Transposing becomes especially useful in operations involving orthogonal matrices since their transpose is also their inverse.

Matrix Inverse

The matrix inverse is like the "reverse" of a matrix. It's a special matrix that, when multiplied with the original matrix, yields an identity matrix. Not every matrix has an inverse, but when a matrix does, this inverse is unique.
Here’s what you need to know:

• For a matrix \( A \), its inverse is denoted as \( A^{-1} \).
• If a matrix is square (same number of rows and columns), it might have an inverse.
• The equation \( AA^{-1} = I \) holds true, where \( I \) is the identity matrix.

Calculating the inverse involves specific methods such as Gaussian elimination, adjoint matrix, or using determinant-based formulae. In the context of orthogonal matrices, the beauty is that the inverse is simply the transpose, which is much easier to compute.

Identity Matrix

The identity matrix is the building block for understanding matrices in linear algebra. It’s a square matrix filled with 0’s except for the diagonal, which is full of 1’s. Imagine it as the matrix equivalent of the number 1, because when any matrix is multiplied by the identity matrix, the original matrix remains unchanged.
Some essential features include:

• An identity matrix is denoted by \( I \), often with a subscript to show the size, like \( I_3 \) for a 3x3 identity matrix.
• The equation \( AI = IA = A \) holds for any matrix \( A \) that can be multiplied by \( I \).

When working with matrices in equations, the identity matrix plays a crucial role, especially when verifying properties of orthogonal matrices or confirming the result of a matrix and its inverse.