Matrix Scalar Product

Why

We have seen that the matrices are a vector space. Are they an inner product space?

Definition

The matrix scalar product of $A \in \R ^{n \times k}$ and $B \in \R ^{n \times k}$ is the following product

\[ \sum_{i = 1}^{n} \sum_{j = 1}^{k} a_{ij}b_{ij}. \]

Using the matrix trace, we can denote this as $\tr A^\top B$. Some authors call this the Euclidean matrix scalar product, matrix inner product or Frobenius inner product.

The matrix scalar product is an inner product.

For example, symmetry of the product is a consequence of the fact that a square matrix and its transpose have identical traces. Commutativity of the trace yields $\tr \transpose{A} B = \tr B\transpose{A}$, where the LHS is the scalar product of $\transpose{B}$ and $\transpose{A}$. In other words, transposition “preserves” the matrix scalar product.

With this inner product, $\R ^{n \times k}$ is a Euclidean vector space (see Inner procuts) of dimension $nk$. For the case of $k = 1$, we recover a model¹ for the usual space $\R ^n$.

Notation

We commonly denote the matrix inner product by $\langle A, B \rangle$.

Induced norm

The matrix inner product induces a norm in the usual way. This norm is sometimes called the matrix-vector norm (or Frobenius norm) and is often denoted for a matrix $A \in \R ^{m \times n}$ by $\norm{A}_F$.

Future editions will define this term. ↩︎