Matrix Terminology

Identity, diagonal, symmetric, orthogonal, transpose, trace — the vocabulary.

The idea

A matrix is a rectangular array of numbers arranged in rows and columns. Its size is written $m \times n$, meaning $m$ rows by $n$ columns, and the entry in row $i$ and column $j$ is written $a_{ij}$, with the row index first.

Most of the standard names are conditions on the entries. A matrix is square when $m = n$; diagonal when every entry off the main diagonal (the entries $a_{ii}$) is zero; upper triangular when every entry below the main diagonal is zero. The identity $I_n$ is the $n \times n$ diagonal matrix whose diagonal entries are all $1$.

The transpose $A^{\top}$ is the $n \times m$ matrix whose $(i, j)$ entry is $a_{ji}$: it reflects the array across the main diagonal. Conditions written with it name whole families: a matrix with $A^{\top} = A$ is symmetric, one with $A^{\top} = -A$ is skew-symmetric, and a square $Q$ with $Q^{\top} Q = I$ is orthogonal. Finally, the trace of a square matrix is the sum of its diagonal entries, $\mathrm{tr}(A) = \sum_i a_{ii}$.

Ways to work on it

• Walkthrough. Match each definition to the name.
• Practice. Pick the term that matches the property.
• Hardest. Combine the definitions to deduce identities about transposes, traces, and symmetry.

Not sure where to start? Take the ten-question placement test.