# Intuition & use of M*M^T product of matrix & its transpose?

1. Jan 28, 2015

### NotASmurf

Hi all, i've occasionly seen people multiply a matrix by its transpose, what is the use and intuition of the product? Any help appreciated.

2. Jan 29, 2015

### Staff: Mentor

Were these orthogonal matrices, where M-1 = MT?

3. Jan 29, 2015

### NotASmurf

In one case yes, What would that mean intuition wise?

4. Jan 29, 2015

### Staff: Mentor

In that case, its use is obvious, as M MT = I. Otherwise, it depends on the context.

5. Jan 29, 2015

### NotASmurf

Was in stats, with covarience matrices.

6. Jan 29, 2015

### Fredrik

Staff Emeritus
These products show up when inner products are involved. For example, if you write elements of $\mathbb R^n$ as n×1 matrices, you can write $x\cdot y=x^Ty$. From this you get results like $(Mx)\cdot(My)=(Mx)^T(My)=x^TM^TMy$.

7. Jan 30, 2015

### homeomorphic

The column vectors are orthonormal. Multiplying by the transpose makes you dot all the column vectors together with each other to get each entry of the product, which it the identity matrix.