Orthogonal matrix

This article defines a property that can be evaluated for a square matrix with entries over the field of real numbers. In other words, given a square matrix (a matrix with an equal number of rows and columns) with entries over the field of real numbers, the matrix either satisfies or does not satisfy the property.
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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 $j\neq k\in \{1,2,\dots ,n\}$.

Significance of underlying ring

The definition of orthogonal matrix presented here makes sense over any commutative unital ring. However, it has particular relevance for matrices where the entries are restricted to the reals or a subring of the reals.