The expression $A^T A$, where $A$ is a matrix and $A^T$ is its transpose, has several important properties, depending on the nature of $A$. Let’s break it down:

1. General Case:

   • $A^T A$ is always a square matrix (even if $A$ is not square).
   • It is a symmetric matrix, meaning $(A^T A)^T = A^T A$.

2. Rank and Positive Semidefiniteness:

   • $A^T A$ is positive semidefinite, meaning all its eigenvalues are non-negative.
   • If $A$ has full column rank (its columns are linearly independent), $A^T A$ is positive definite (eigenvalues are positive).

3. Dimensionality:

   • If $A$ is an $m \times n$ matrix, $A^T A$ will be an $n \times n$ matrix.

4. Geometric Meaning:

   • $A^T A$ often arises in the context of inner products or projections in linear algebra, such as in least-squares problems, where it represents the Gram matrix of $A$’s column vectors.

5. Special Cases:

   • If $A$ is orthogonal ($A^T A = I$), $A^T A$ simplifies to the identity matrix $I$.
   • If $A$ is diagonal or symmetric, $A^T A$ preserves certain structural properties.

Would you like further examples, computations, or geometric interpretations of $A^T A$?

Related Questions:

1. What is the physical significance of $A^T A$ in applications like least squares or machine learning?
2. How does $A^T A$ differ when $A$ is rectangular versus square?
3. Can $A^T A$ be singular, and under what conditions?
4. What does $A^T A$ look like if $A$ contains orthogonal vectors as columns?
5. How do eigenvalues of $A^T A$ relate to singular values of $A$?

Tip:

When working with $A^T A$, computing its eigenvalues or singular values can reveal insights about the transformations encoded by $A$.