How can I indicate that empty elements of a matrix are zero?

Question

I have the following transition matrix in an academic conference paper:

$$\mathbf{P}=\begin{bmatrix} 1-p & p & & & & \\ & 1-p & p & & & \\ & & & \ddots & & \\ & & & & 1-p & p\\ & & & & & 1 \end{bmatrix}$$

What is the best way to indicate that the empty elements are zero? I could also add a description in the body of the text. I considered something like "the off-diagonal elements are zero", but there are nonzero elements off of the diagonal so this wouldn't be entirely accurate.

Answer 1

• One simple way: mention $P$ is bidiagonal.

  Let $\mathbf{P}$ be the $n\times n$ bidiagonal matrix $$\mathbf{P}=\begin{bmatrix} 1-p & p & & & & \\ & 1-p & p & & & \\ & & & \ddots & & \\ & & & & 1-p & p\\ & & & & & 1 \end{bmatrix}$$

• Another: write it (semi) explicitly by adding a big zero to indicate blocks identically zero (this is quite standard).

  Let $\mathbf{P}$ be the $n\times n$ matrix $$\mathbf{P}=\begin{bmatrix} 1-p & p & & & & \\ & 1-p & p & & \Large 0 & \\ & & & \ddots & & \\ & \Large 0 & & & 1-p & p\\ & & & & & 1 \end{bmatrix}$$

• A third: combine both.

  Let $\mathbf{P}$ be the $n\times n$ bidiagonal matrix $$\mathbf{P}=\begin{bmatrix} 1-p & p & & & & \\ & 1-p & p & & \Large 0 & \\ & & & \ddots & & \\ & \Large 0 & & & 1-p & p\\ & & & & & 1 \end{bmatrix}$$

Note: to be even more specific, you can write "upper bidiagonal."