Explanation of notation of dot above plus $\dotplus$ for matrices

Question

I am studying Y Kubota's article "A note on holomorphic imbeddings of the classical Cartan domanis into the unit ball". There is a phrase in the article

Considering a skew-symmetric matrix $Z$ of order $q$ such that $$Z=\left(\begin{array}{cc} 0 & a_1\\ -a_1&0 \end{array}\right)\dotplus\ldots\dotplus \left(\begin{array}{cc} 0 & a_m\\ -a_m&0 \end{array}\right),$$ where q=2m and $a_1,\ldots, a_m\in{\mathbb{C}}$.

Clearly, the expressions on the right are $2\times 2$ matrices and the expression on left is a matrix of order $2m$, but what does the symbol $\dotplus$ mean.

I found similar notation here but it is for product of algebras. Can someone help me understand what it means, if it is a standard notation or what it means.

Answer 1

The 1980 article "A Note on Holomorphic Imbeddings of the Classical Cartan Domains into the Unit Ball" by Yoshihisa Kubota uses $\dotplus$ and cites the 1944 article "On the Theory of Automorphic Functions of a Matrix Variable I-Geometrical Basis" by Loo-Keng Hua, who writes in Theorem 7:

Let $Z$ be a non-singular skew-symmetric matrix; then we have a unitary matrix $U$ such that $$UZU'=\begin{pmatrix}0&d_1\\-d_1&0\end{pmatrix}\dotplus\cdots\dotplus\begin{pmatrix}0&d_{n/2}\\-d_{n/2}&0\end{pmatrix}\text{,}$$ where $d_1^2,d_1^2,\cdots,d_{n/2}^2,d_{n/2}^2$ are the characteristic roots of $-Z\overline{Z}$.

For comparison, the spectral theory section of the English Wikipedia page for skew-symmetric matrix says

$$\Sigma = \begin{bmatrix} \begin{matrix}0 & \lambda_1 \\ -\lambda_1 & 0\end{matrix} & 0 & \cdots & 0 \\ 0 & \begin{matrix}0 & \lambda_2 \\ -\lambda_2 & 0\end{matrix} & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & \begin{matrix}0 & \lambda_r\\ -\lambda_r & 0\end{matrix} \\ & & & & \begin{matrix}0 \\ & \ddots \\ & & 0 \end{matrix} \end{bmatrix}$$...every complex skew-symmetric matrix can be written in the form $A=U\Sigma U^{\mathrm{T}}$ where $U$ is unitary and $\Sigma$ has the block-diagonal form given above with $\lambda_k$ still real positive-definite.

With this context, I am fairly confident that $\dotplus$ as used by Kubota denotes the direct sum of matrices, which produces a block form as exemplified above and guessed by gary in a comment.