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Chapter 5: Problem 25

Prove that every permutation matrix is orthogonal.

Short Answer

Expert verified
Permutation matrices are orthogonal because their transpose equals their inverse.

Step by step solution

01

Define a Permutation Matrix

A permutation matrix is a square matrix obtained by permuting the rows of an identity matrix. In other words, if we take an identity matrix and rearrange its rows, the resulting matrix is a permutation matrix. Each row and each column has exactly one entry of 1 and all remaining entries are 0.
02

Recall the Definition of an Orthogonal Matrix

A matrix is orthogonal if its inverse is equal to its transpose. Mathematically, a matrix \( P \) is orthogonal if \( P^T P = I \) where \( I \) is the identity matrix.
03

Determine the Transpose of the Permutation Matrix

Take the permutation matrix, \( P \), and calculate its transpose, \( P^T \). The transpose of a permutation matrix is obtained by swapping its rows and columns, essentially permuting columns of the original identity matrix, similar to how rows were permuted to form \( P \).
04

Multiply the Transpose with the Original Matrix

Now, calculate \( P^T P \). Since both \( P^T \) and \( P \) have exactly one "1" per row and per column, all off-diagonal elements after multiplication will be zeros and the diagonal elements will be "1", because if two rows (or columns) of \( P \) are different in \( P^T \), then their dot product is zero.
05

Conclude with the Identity Matrix

Upon calculating \( P^T P \), you will find that the result is the identity matrix \( I \). This confirms that for permutation matrices \( P \), \( P^T P = I \), thus proving that permutation matrices are orthogonal.

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Key Concepts

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

Permutation Matrix
A permutation matrix is a special type of square matrix obtained by reordering the rows of an identity matrix. This means each row and column in a permutation matrix uniquely contains a single '1', with all other elements being '0'.
This characteristic maintains the orthogonality properties.
  • It is used to rearrange or permute other matrices or vectors.
  • Useful in algorithms and numerical computations, where data rearrangement is needed.
The beauty of permutation matrices is in how straightforwardly they embody change while preserving some fundamental characteristics of matrices like being orthogonal. Understanding permutation matrices deeply involves seeing them as the blueprint for how elements in a mathematical setting can be systematically reorganized without losing core properties.
Identity Matrix
The identity matrix is a simple yet powerful concept in linear algebra. It is a special square matrix with all elements on the main diagonal as 1, and all other elements as 0. Thus, multiplying any matrix by the identity matrix leaves it unchanged.
  • For an identity matrix of size \( n \times n \), it is denoted by \( I_n \).
  • The identity matrix acts as the multiplicative identity for matrices, much like the number 1 is for numbers.
This matrix serves as a foundational building block in linear transformations, acting as a neutral element in matrix operations, ensuring results remain unaffected when used in multiplication.
Matrix Transpose
Transposing a matrix involves switching its rows with its columns. The transpose of a matrix \( A \) is represented as \( A^T \). In a permutation matrix, transposing has a unique property of preserving the orthogonal nature.
  • The main diagonal remains unchanged during a transpose.
  • The operation is simple: the element in the i-th row and j-th column in the original matrix becomes the element in the j-th row and i-th column in the transposed matrix.
Transposing is central in various calculations, especially when determining if a matrix is orthogonal, such as with permutation matrices.
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra where the rows of the first matrix are multiplied with the columns of the second matrix. The result is a new matrix.
  • For multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix.
  • The element at position (i, j) in the resulting matrix is the dot product of the i-th row of the first matrix and the j-th column of the second.
In permutation matrices, multiplying the transpose by the matrix itself results in the identity matrix, which is a key condition for a matrix to be orthogonal. This property showcases the unique symmetry and stability found in these matrices.

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