In which cases is the inverse matrix equal to the transpose?

Question

As said in the title, in which cases an invertible matrix is equal to the transpose? When is this: $ A^{-1} = A^{T} $ true?

If the matrix A is orthogonal?

Thank you!

Do you want to ask about invertible matrices that are equal to their transposes, or matrices $A$ such that $A^{-1}=A^T$? These are not the same thing.
Well I want to find out if this expression: $Q^{T}AQ$ is equal to this: $QAQ^{-1}$, where Q is an orthogonal matrix and A is a symmetric one. @ChrisEagle
@talmid: So there is not a problem that the transpose is on the left at the 1st expression and the invertible is on the right at the 2nd? Thank you for your reply! :)
@Chris : Could you change your subject line if it's not what you mean? If you're asking for which matrices $A$ is $A^T$ the same as $A^{-1}$, then you're not asking when an invertible matrix is equal to its transpose. That's a different question and has a different answer. Could you alter your subject line to reflect what you're trying to ask? It could say "In which cases is the inverse of a matrix equal to the transpose?". There's something to be said for being comprehensible.

robjohn · Accepted Answer · 2012-06-10 23:36:33Z

up vote 6 down vote accepted

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 is an orthogonal matrix.

answered Jun 10 '12 at 23:36

robjohn♦
59.5k352153

talmid · Answer 2 · 2012-06-11 22:25:39Z

up vote 2 down vote

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

edited Jun 11 '12 at 22:25

answered Jun 10 '12 at 23:09

talmid
1,24829

2 Answers