Let A be an m ×n matrix whose columns are nonzero, mutually...

Let $A$ be an $m \times n$ matrix whose columns are nonzero, mutually orthogonal vectors in $\mathbb{R}^m$. (a) Explain why $m \geq n$. (b) Prove that $A^T A$ is a diagonal matrix. What are the diagonal entries? (c) Is $\bar{A} A^T$ diagonal?

Key Concepts

Orthogonal Vectors

Orthogonal vectors in a Euclidean space have a zero inner product with each other. This concept is fundamental because when nonzero vectors are mutually orthogonal, they are automatically linearly independent, which is a key property used in many areas of linear algebra including matrix factorizations and projections.

Matrix Dimensions and Linear Independence

The concept that the number of mutually orthogonal nonzero vectors cannot exceed the dimension of the ambient space is crucial. This means that for an m × n matrix with columns in ?^m, if the columns are mutually orthogonal (and nonzero), then n must be less than or equal to m. This guarantees that the columns form a set of linearly independent vectors.

Matrix Products and Diagonality

Understanding how matrix multiplication interacts with vector orthogonality is important. The product A?A, where A's columns are mutually orthogonal, results in a diagonal matrix. This occurs because the off-diagonal entries become the dot products between distinct columns, which are zero, and the diagonal entries are the norms squared of the columns.

Properties of Matrix Products Involving Transposes

While the product A?A is diagonal due to the column orthogonality, analyzing products such as A A? (or conjugate versions) involves different considerations. Such products aggregate information about both the column and row space, and in general, need not be diagonal even when the columns of A are orthogonal.

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