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.

I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition of transpose. Suppose $A_{m \times n}$, and A transform a vector from $\Bbb R^n$ to $\Bbb R^m$. But $A^T$ transform a vector from $\Bbb R^m$ to $\Bbb R^n$. What's the relationship between them? Could anyone please explain the relationship between $A^T$,$A$,the inner product and symmetric matrix. I think there would be a intuition explaination.

share|improve this question
See: math.stackexchange.com/questions/37398/… –  Casteels Sep 30 '13 at 14:53
add comment

2 Answers

Well, $A^T$ is the adjoint matrix of $A$ with respect to the ordinary inner products, i.e. $A^T$ is the only linear mapping $B$ such that $$\langle Av,w\rangle = \langle v,Bw\rangle$$ for all $v\in\Bbb R^n$ and $w\in\Bbb R^m$. You can easily see it if you verify it on the standard bases, noting that $\langle u,e_i\rangle$ gives the $i$th coordinate of $u$.

share|improve this answer
thank you. But how to use this conclusion? Is there any geometry explaination? –  maple Jan 10 '13 at 11:58
add comment

One aspect of this to consider is that the transpose lets you do the same thing in different ways. The regular matrix gets multiplied by a column on the right to give your answer as a column. The transpose gets multiplied by a row on the left to give a row as the answer.

Is one better than the other? Not really, they're equivalent.

share|improve this answer
add comment

Your Answer


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.