Question

If \(P\) is a triangular orthogonal matrix, show that \(P\) is diagonal and that all diagonal entries are 1 or -1.

Short answer

Triangular orthogonal matrices are diagonal, and diagonal entries are 1 or -1.

Step-by-step solution

Define a Triangular Orthogonal Matrix

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors, which implies that the matrix satisfies \( P^T P = I \). A triangular matrix, on the other hand, is a type of square matrix that has all zeros either above or below its main diagonal.

Apply Orthogonality to a Triangular Matrix

Since \( P \) is orthogonal, for any column vector \( p_i \, (i=1, \, 2, \, ..., \, n) \) in \( P \), we have \( \|p_i\| = 1 \). As \( P \) is also triangular, either upper or lower, the non-zero components must only lie on the diagonal. For an orthogonal matrix, the eigenvalues must have a magnitude of 1.

Consider Properties of Triangular Matrix

A triangular matrix implies all elements below (or above) the diagonal are zeroes. When combining this feature with orthogonality, the only possible unit vectors for non-zero components are on the diagonal.

Check Diagonal Entries for Magnitude One

An orthogonal matrix's determinant is either 1 or -1. For a triangular matrix, the determinant equals the product of the diagonal elements, implying each diagonal element of \( P \) must be \( \pm 1 \) to satisfy both being orthogonal and triangular.

Conclude with Diagonal Elements' Values

Since the triangular part is zero and \( P \) is orthogonal, all diagonal elements must be either 1 or -1 to ensure the determinant is \( \pm 1 \). This proves any triangular orthogonal matrix is also diagonal with diagonal elements 1 or -1.

Key concepts

Triangular Matrices

Triangular matrices are a special type of square matrix. They have a clear pattern where all elements are zero either above or below the main diagonal. This specific arrangement can be:

• Upper triangular, where all elements below the diagonal are zero.
• Lower triangular, where all elements above the diagonal are zero.

These matrices are important in matrix algebra due to their simplicity in computations. For instance, the determinant of a triangular matrix is simply the product of its diagonal elements. This characteristic greatly reduces the complexity of determinant calculations.
Additionally, triangular matrices are fundamental in linear equations and eigenvalue problems because they are easier to manipulate than general matrices. Their structure can simplify solving systems of linear equations via methods such as forward or backward substitution.

Matrix Orthogonality

Matrix orthogonality is a concept where a matrix, say matrix \(P\), has orthogonal rows and columns. This means each row and each column has a dot product of zero with every other row and column, respectively.
More formally, an orthogonal matrix satisfies \( P^T P = I \), where \( P^T \) is the transpose of \( P \), and \( I \) is the identity matrix, implying that the transpose of \( P \) is also its inverse.
Some critical properties of orthogonal matrices include:

• Preservation of vector lengths: Multiplying a vector by an orthogonal matrix does not change its length.
• Determinants: The determinant of an orthogonal matrix is always \(\pm 1\), which relates to flipping or preserving orientations.

These properties make orthogonal matrices extremely useful in applications like computer graphics and numerical stability in computations.

Eigenvalues and Determinants

Eigenvalues and determinants are crucial concepts in understanding the behavior of matrices. To start with, eigenvalues are scalars that provide significant insight into the matrix's properties and are identified from the equation:\[ A \mathbf{v} = \lambda \mathbf{v} \]Here, \(A\) is the matrix, \(\lambda\) represents the eigenvalues, and \(\mathbf{v}\) refers to the eigenvectors.
In the context of orthogonal matrices, all eigenvalues have an absolute value of 1. This occurs because orthogonal transformations do not change the length of vectors.
The determinant of a matrix is a scalar value that provides essential information about the matrix, such as its invertibility. For triangular matrices, the determinant is simply the product of its diagonal elements. Therefore, the determinant of an orthogonal triangular matrix \(P\), given all diagonal entries are either 1 or -1, will be \(\pm 1\).
This connection between eigenvalues and determinants showcases that if a matrix is both orthogonal and triangular, it will necessarily be diagonal with eigenvalues of magnitude 1.