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

[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.

 

[DrClaude]

Were these orthogonal matrices, where M^-1 = M^T?

 

[NotASmurf]

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

 

[DrClaude]

In one case yes,

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

 

[NotASmurf]

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

Was in stats, with covarience matrices.

 

[Fredrik]

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.

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 $(Mx)\cdot(My)=(Mx)^T(My)=x^TM^TMy$.

 

[homeomorphic]

What would that mean intuition wise?

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.