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.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If A is any real $m \times n$ matrix, then $A^TA$ is called the Gram matrix of $A$. Show that the Gram matrix of $A$ is always symmetric with nonnegative diagonal elements.

I have tried several matrices to prove that this is true but im not sure how I go about showing for a general matrix that this always holds true. Please help.

share|cite|improve this question
What is the $(i,j)$ entry of $A^T$? What is the $(i,j)$ entry of the matrix product $BA$? What is the $(i,j)$ entry of the matrix product $A^TA$? – user108903 Jan 21 '13 at 19:15


  1. Prove that the Gram matrix is always symmetric: What does symmetry have to do with transposes? What happens when you transpose $A^TA$?

  2. Prove that the diagonal elements are nonnegative: Let $a_{ij}$ be the $(i,j)$ element of A. Can you find an expression for the $k$-th diagonal element of $A^TA$ in terms of the $a_{ij}$?

share|cite|improve this answer
@user56881 You're making the symmetry argument much harder than you need to. What is $(A^TA)^T$? – Jonathan Christensen Jan 21 '13 at 19:36

Some hints: for first part, use $(AB)^T=B^TA^T$

for second part, use $a_{ii}=e_i^TA^TAe_i=(Ae_i)^T(Ae_i)=||Ae_i||^2\geq0$. ($e_i$ is $i^{th}$ column of identity matrix)

share|cite|improve this answer
@user56881 You sort of got it. We don't need to show that $A^T = A$, we need to show that $G^T = G$, which is what you showed. $A$ may not even be square, so it's certainly not necessarily symmetric. – Jonathan Christensen Jan 21 '13 at 20:30

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.