Orthogonal matrix

This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices

Definition

Verbal definition

A square matrix is termed an orthogonal matrix if its matrix transpose equals its inverse matrix. In symbols, a $n\times n$ matrix $A$ is termed an orthogonal matrix if $A^{T}=A^{-1}$, or equivalently, $AA^{T}=I_{n}$, the $n\times n$ identity matrix.

Algebraic definition

Suppose $n$ is a positive integer. A $n\times n$ square matrix $A=(a_{ij})_{1\leq i\leq n,1\leq j\leq n}$ is termed an orthogonal matrix if the following hold:

• The rows have unit norm: $\sum _{j=1}^{n}a_{ij}^{2}=1$ for any $i\in \{1,2,\dots ,n\}$
• Distinct rows are orthogonal: $\sum _{j=1}^{n}a_{ij}a_{kj}=0$ for any $i\neq k\in \{1,2,\dots ,n\}$.

The above conditions are framed in terms of the rows, but an equivalent formulation in terms of the columns works (this is not a priori obvious, and follows from the involutive nature of the transpose and inverse operations). Explicitly:

• The rows have unit norm: $\sum _{i=1}^{n}a_{ij}^{2}=1$ for any $j\in \{1,2,\dots ,n\}$
• Distinct rows are orthogonal: $\sum _{i=1}^{n}a_{ij}a_{ik}=0$ for any $j\neq k\in \{1,2,\dots ,n\}$.