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While thinking about what it means for observables to be simultaneously measurable in quantum mechanics I came up with the following concepts, which I will call "linearly indexed" versions of standard linear algebra concepts.

  • A $W$-indexed linear map $L: V \to V$ is a linear map $W \otimes V \to W \otimes V$.
  • A $W$-indexed eigenspace for $L$ is a subspace $V'$ of $V$ such that $L = \lambda \otimes I$ on $W \otimes V'$.
  • $\lambda : W \to W$ is called the $W$-indexed eigenvalue of $L$ for the $W$-indexed eigenspace $V'$.

When the spaces are over $\mathbb{C}$, say, and $W = \mathbb{C}$, we just recover the normal definition of eigenspace and eigenvalue $\lambda \in \mathbb{C}$.

It seems to me that these concepts allow us to talk about multidimensional observables of a quantum mechanical system in a direct way. An observable on a space of states $V$ is a Hermitian $W$-indexed linear map $V \to V$, and the values it can take are its $W$-indexed eigenvalues, which should to be Hermitian operators $W \to W$.

Is this something that is known and studied, or is this a red herring?

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The definition of $W$-indexing requires some further clarification. As stated right now, all three properties are satisfied by the classic Kronecker product.

To illustrate, suppose $V=\mathbb{C}^n$, $W=\mathbb{C}^m$. For argument's sake, let us assume that $L$ has simple eigenvalues, such that a diagonalization exists $L=X\Lambda X^{-1}$, and each eigenspace is dimension-1.

Then for any $w\in W$, $v\in V$, and any bijective linear map $D:W\to W$ which is also an invertible $m\times m$ matrix, we have

  • The $W$-indexed linear map $L$ is $\hat{L}=D\otimes L$. The domain is $w \otimes v \in W\times V$ and the range is $\hat{L}(w\otimes v)=(Dw)\otimes (Lv)\in W\times V$, so long as $L$ is nonsingular.
  • For any eigenpair $\lambda_i, x_i$ of $L$, we have $\hat{L}(w \otimes x_i)= (Dw) \otimes (Lx_i) = (Dw) \otimes (\lambda_i x_i) = (\lambda_i D\otimes I_n)(w \otimes x_i).$ Again, so long as $L$ is nonsingular, we have $\lambda_i\ne0$, so the range is $(\lambda_i D\otimes I_n)(w \otimes x_i) \in W \times x_i$
  • In the example above, the matrix $\lambda_i D$ is a $W\to W$ mapping that fits the definition of a $W$-indexed eigenvalue.

Of course if we set $D=1$, then we recover the usual eigenvalues and eigenvectors, as per stated.

Given any arbitrary $A$, $B$, the eigenvalues, eigenvectors, Jordan form, singular decomposition, etc, of $A\otimes B$ can all be expressed using properties of $A$ and $B$, and some basic linear algebra identities. See the wiki page.

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