49

In which cases is the inverse of a matrix equal to its transpose, that is, when do we have $A^{-1} = A^{T}$? Is it when $A$ is orthogonal?

Chris
  • 777
  • 2
  • 7
  • 14

2 Answers2

62

If $A^{-1}=A^T$, then $A^TA=I$. This means that each column has unit length and is perpendicular to every other column. That means it is an orthonormal matrix.

azureai
  • 1,582
  • 12
  • 29
robjohn
  • 326,069
  • 34
  • 421
  • 800
  • 2
    at the risk of reviving a dodgy question, may I ask "why" the geometric interpretation of *orthogonal matrix* is equivalent to the algebraic definition you gave? I know the property, but I don't understand it. – bright-star Dec 27 '13 at 08:22
  • 4
    @TrevorAlexander: Think of $A$ as an arrangement of $n$ columns (each $n$ elements tall). Then the $(i,j)$ element of $A^TA$ is the dot product of the $i^\text{th}$ and $j^\text{th}$ columns of $A$ since the $i^\text{th}$ row of $A^T$ is the $i^\text{th}$ column of $A$. – robjohn Dec 27 '13 at 10:03
  • 1
    could you give me confidence that this is actually an "if and only if"? I mean that **both** directions hold: $A^{-1} = A^\top \Leftrightarrow A^\top A = I$ – Milla Well Mar 25 '14 at 17:20
  • 4
    @MillaWell: $A^{-1}=A^T\implies A^TA=I$: Multiply both sides on the right by $A$. $A^TA=I\implies A^{-1}=A^T$: By definition. – robjohn Mar 25 '14 at 18:40
8

You're right. This is the definition of orthogonal matrix.

talmid
  • 1,567
  • 8
  • 10