Lagmental Vicfred

Kronecker Products Multiply Blockwise by Vicfred

tags: mathematics, multilinear algebra, tensor products, matrix functions

The tensor product of two matrix maps is represented by the Kronecker product. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

The mathematical object

Tensor and exterior powers turn multilinear behavior into linear maps. For finite-dimensional \(V\), the spaces \(V^{\otimes k}\) and \(\bigwedge^kV\) carry induced actions of every \(T\in\operatorname{End}(V)\).

$$ A\otimes B=(a_{ij}B)_{ij} $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ (A\otimes B)(C\otimes D)=AC\otimes BD $$

One explicit computation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \begin{pmatrix}a&b\\c&d\end{pmatrix}\otimes B=\begin{pmatrix}aB&bB\\cB&dB\end{pmatrix},\qquad\det(A\otimes B)=\det(A)^m\det(B)^n $$

Why the identity matters

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad A\otimes B=(a_{ij}B)_{ij},\\[5pt] \mathsf{C}\;&:\quad (A\otimes B)(C\otimes D)=AC\otimes BD. \end{aligned} $$

Where it can fail

Tensor coordinates depend on a basis even when the tensor does not. Index notation is safe only when contraction rules and variance are clear.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] (A\otimes B)(C\otimes D)=AC\otimes BD \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Sun 26 March 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.