Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
2  
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. –  Chris Eagle Jun 10 '12 at 23:10
    
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 –  Chris Jun 10 '12 at 23:13
    
@Chris It's equal to $Q^{-1}AQ$. –  talmid Jun 10 '12 at 23:16
    
@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 Jun 10 '12 at 23:26
1  
@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. –  Michael Hardy Jun 10 '12 at 23:27

2 Answers 2

up vote 11 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.

share|improve this answer
    
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. –  Trevor Alexander Dec 27 '13 at 8:22
1  
@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
    
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 at 17:20
    
@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 at 18:40

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.