Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

What does it mean if a matrix is multiplied by its transpose? Informally, it seems like $A^TA$ boils a matrix down to its essentials, but can this operation somehow be understood "intuitively" (e.g. through a geometric interpretation)?

share|cite|improve this question

To elaborate on user9325's answer, the standard inner product of two vectors $x, y$ is given by $x^T y$. If we change coordinates $x \mapsto Ax', y \mapsto Ay'$, then the inner product becomes $x^T y \mapsto x'^T (A^T A) y'$, so the matrix $A^T A$ encodes the coefficients of the new inner product.

share|cite|improve this answer

I will assume that your matrix is real.

1) If $A$ encodes a change of basis, you can ask what happens to the standard inner product of two vectors. The matrix above gives you the formula for this product in the new basis.

2) The eigenvalues of your product are the singular values of the matrix (in particular, it does not have to be a square matrix for this interpretation). Note that the product "makes the matrix square and symmetric".


(which also explains that this gives the length of the semi-axes of an ellipse in a special case)

share|cite|improve this answer

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.