Let A be an m ×n matrix. (a) Explain why the product L=A A^T...

Let $A$ be an $m \times n$ matrix. (a) Explain why the product $L=A A^T$ is a Gram matrix.
(b) Show that, even though they may be of different sizes, both Gram matrices $K=A^T A$ and $L=A A^T$ have the same rank. (c) Under what conditions are both $K$ and $L$ positive definite?

Key Concepts

Gram Matrix

A Gram matrix is formed from a set of vectors by taking the inner products between all pairs of vectors. When you express a matrix A whose rows (or columns) represent the vectors, the matrix product AA? computes the inner products between the rows of A. This makes AA? a Gram matrix because each entry represents the dot product of two vectors, encapsulating the geometric relationships among them.

Rank Equivalence

The rank of a matrix is the number of linearly independent rows (or columns) it contains. For any matrix A, the products A?A and AA? have the same nonzero eigenvalues as A. This is due to properties of singular value decomposition, where the nonzero singular values appear in both products. Therefore, even though A?A and AA? are of different sizes, they share the same rank, which is determined by the rank of A.

Positive Definiteness Conditions

A Gram matrix is always symmetric and positive semidefinite because it is constructed from dot products. For a Gram matrix to be positive definite, the set of vectors used must be linearly independent, ensuring that the matrix is invertible. In the context of A?A (an n×n matrix) and AA? (an m×m matrix), positive definiteness is achieved when A has full column rank in the first case and full row rank in the second case. Thus, both matrices are positive definite if and only if A is of full rank in the corresponding dimension.

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