Question

a. Let \(A\) be an \(m \times n\) matrix. Show that the following are equivalent. i. \(A\) has orthogonal rows. ii. \(A\) can be factored as \(A=D P,\) where \(D\) is invertible and diagonal and \(P\) has orthonormal rows. iii. \(A A^{T}\) is an invertible, diagonal matrix. b. Show that an \(n \times n\) matrix \(A\) has orthogonal rows if and only if \(A\) can be factored as \(A=D P,\) where \(P\) is orthogonal and \(D\) is diagonal and invertible.

Short answer

a. (i), (ii), and (iii) are equivalent. b. An \( n \times n \) orthogonal matrix can be factored as \( A = DP \) with \( P \) orthogonal and \( D \) diagonal and invertible.

Step-by-step solution

Definition and Basic Understanding

First, recall the definitions. An orthogonal row set of a matrix means that the dot product of different rows is zero. An orthonormal row set means that rows are orthogonal and each row is a unit vector.

Prove (i) implies (ii)

Assume that matrix A has orthogonal rows. We seek to express it as \( A = DP \). Let each row of A be a scalar multiple of an orthonormal vector. Then, construct a diagonal matrix D with these scalar multiples along the diagonal and a matrix P consisting of the corresponding orthonormal vectors as rows. Thus, \( A = DP \) where D is invertible and diagonal and P has orthonormal rows.

Prove (ii) implies (iii)

Assume that \( A = DP \) where D is invertible and diagonal, and P has orthonormal rows. Calculate \( A A^{T} = (DP)(DP)^T = DPP^TD^T = D I D^T = D D^T \). Since D is invertible and diagonal, \( D D^T \) is a diagonal matrix with non-zero entries, hence it is invertible.

Prove (iii) implies (i)

Assume that \( A A^T \) is an invertible diagonal matrix. This implies that each diagonal entry represents the squared norm of the corresponding row. Since none of these can be zero (to maintain invertibility), all rows are non-zero vectors. As off-diagonal entries are zero, distinct rows are orthogonal.

Special Case for n x n Matrix

Given \( A \) is an \( n \times n \) matrix with orthogonal rows. From \( A A^T \) being diagonal and invertible, we have \( A = DP \) by previous logic with P being orthogonal. Thus, an \( n \times n \) matrix with orthogonal rows can be factored as \( A = DP \), where \( P \) is orthogonal and \( D \) is diagonal and invertible.

Key concepts

Matrix Factorizations

Matrix factorization is a powerful tool in linear algebra, allowing us to decompose a complex matrix into simpler, more manageable pieces. In our exercise, we explore a specific factorization: the decomposition of a matrix into a product of a diagonal matrix and a matrix with orthonormal rows.

To factor a matrix, we typically look for a form where it can be expressed as a product of two or more matrices, each possessing specific desirable properties. The goal is to facilitate easier computations or to reveal certain characteristics about the original matrix. For instance, representing matrix \( A \) as \( A = DP \) helps identify relationships between the rows of \( A \) and provides insights into its structural properties.

This process not only simplifies many operations but also opens up pathways to deeper understanding and further analysis, such as checking for orthonormality or invertibility.

Diagonal Matrices

Diagonal matrices have a distinctive form where only the diagonal elements can be non-zero, while all off-diagonal elements are zero. In the context of our exercise, diagonal matrices feature prominently in matrix factorizations. They simplify complex operations like matrix multiplication and determination of powers.

A real advantage of diagonal matrices is their simplicity in calculations, making them a focal point when discussing matrix operations. For instance, if you have a diagonal matrix \( D \), then \( D^{-1} \) (if exists) is simply another diagonal matrix with reciprocal values of the original diagonal elements, which showcases the importance of invertibility in mathematical computations.

In our matrix factorization, \( D \) must be invertible, meaning none of its diagonal elements are zero. This feature ensures that the transformations represented by \( D \) can be reversed, a critical requirement in many applications.

Orthonormality

Orthonormality is a concept that combines orthogonality with the additional requirement that each vector must have a unit norm (magnitude of 1). When we say a matrix has orthonormal rows, it means that each row is both orthogonal to the others and is a unit vector.

This has significant implications for the matrix's properties and behavior. A matrix with orthonormal rows is particularly nice to work with because it preserves the length of vectors in certain transformations.

Moreover, when a matrix \( P \) has orthonormal rows or columns, it also means \( P^T P = I \), where \( I \) is the identity matrix. This relationship greatly simplifies both theoretical and practical computations, especially involving inner products and projections.

Invertible Matrices

An invertible matrix, often referred to as a non-singular or non-degenerate matrix, is a square matrix that has an inverse. This means that when the matrix is multiplied by its inverse, the result is the identity matrix.

Invertibility is a crucial property because it indicates that the matrix's transformations can be undone or reversed. For a matrix \( D \) in the factorization \( A = DP \) to be invertible, all its diagonal elements must be non-zero as per the diagonal matrix property discussed earlier.

Determining whether a matrix is invertible is essential for ensuring the matrix can be used in further computations like solving systems of equations, making predictions in data science applications, and more. The condition that \( D \) is diagonal and invertible supports robust and reliable calculations.