Matrices

Grid of numbers

A matrix is just a grid of numbers; rows and columns containing values. Matrices can be of any size. Here’s an example of a $3\times 2$ matrix. It is $3$ columns wide and $2$ rows tall, containing the values $1$ through $6$.

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$

When describing the dimensions of a matrix we always specify the number of rows first, then the number of columns. The above is an example of a $3\times 2$ matrix, while below is a matrix containing similar data, but in a $2\times 3$ configuration.

$\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]$

Order matters

For our purposes, we’ll express our matrices in row-major order. This means we read the values just as they are ordered above, starting with the top-most row, reading values from left to right, then proceeding to the next row down and repeating that process. Our choice of row-major order makes reading and writing matrix values more akin to reading and writing in English; easier to type and program here.

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$

$\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right]$

Vectors are slices

While a matrix is a two-dimensional grid of numbers, a vector is more like a slice of numbers—such as a “skinny” matrix that is only one column wide, or a “flat” matrix that is only one row high. Let’s look at some examples of matrices that are simultaneously vectors.

$\left[\begin{array}{cccc}1& 2& 3& 4\end{array}\right]$

$\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]$

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

While the above $2\times 2$ matrix is not a vector, you could say that it contains vectors: Two column vectors or two row vectors. Because vectors are just a type of matrix we can add them, multiply them, and so on—just like any other matrix. Vectors will play a prominent role in defining qubits and expressing the state of a quantum circuit.

Complex numbers

In addition to storing regular numbers, matrices can contain complex numbers. This is both useful and necessary: Qubits are really just a pair of complex numbers that we store in a $1\times 2$ matrix. So, yes—the example matrices above are each larger than a qubit!