matrix inner product

Topics

• linear algebra
• matrices
• inner product

Matrix inner product extends vector inner product to matrices. Treat matrices as stacked column vectors, then compute inner product between these long vectors

Key properties:

• Generalizes dot product to matrices
• Measures “alignment” between two matrices
• Used in matrix norms and optimization problems

Example

Frobenius inner product for $A,B\in {\mathbb{R}}^{m\times n}$:

$$⟨A,B⟩=Tr(A^{T}B)=\sum _{i,j}{A}_{ij}{B}_{ij}$$

Let:

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$$

1. Element-wise multiplication: $A_{ij}{B}_{ij}$
2. Sum all products: $1\times 5+2\times 6+3\times 7+4\times 8=70$

Using trace formula:

$$A^{T}B=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]=\left[\begin{array}{cc}26& 30\\ 38& 44\end{array}\right]$$

Result is same: $Tr(A^{T}B)=26+44=70$

Note

This matches vector inner product when matrices are single column vectors

Related

• matrix multiplication