Kronecker Vectorization
Chapter 10: Matrix Block Partitioning — Vector Kronecker Product
From the book
Chapter 10: Matrix Block Partitioning. In the chapter mind map this icon labels Vector Kronecker Product: $\mathrm{vec}(\mathbf{a}\mathbf{b}^\top) = \mathbf{b} \otimes \mathbf{a}$. The discussion below is excerpted and lightly edited from § Vector Kronecker Product in Mathematics for AI and Machine Learning.
For vectors $\mathbf a \in \mathbb{R}^{m \times 1}$ and $\mathbf b \in \mathbb{R}^{n \times 1}$, the Kronecker product is:
This is equivalent to the outer product $\mathbf a \mathbf b^\top$ when reshaped appropriately.
What this drawing shows
What you see. Represents the identity connecting Kronecker products with vectorized matrix multiplication.
In the mind map. Chapter 10 — Vector Kronecker Product. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 10: Matrix Block Partitioning — see the mind-map node Vector Kronecker Product.