Krdytkyu

Motivation

If $a$ and $b$ are vector, then thinking simply vector 2 norm, $Vert a cdot bVert = Vert bVert Vert aVert cos(a,b) $, we know the difference is simply a ratio between the angle of $a$ and $b$.

More generally, in a Hilbert space, Cauchy inequality holds so
$$|langle a,brangle|^2 le langle a,aranglelangle b,brangle$$
and we know the only when a, b are parallel, the equality is achieved.

Question

Given two square matrix $A$ and $B$,

when does this happen?
$$Vert AB Vert = Vert A Vert Vert B Vert$$

Let's simply assume matrix 2-norm, so $Vert cdot Vert = Vert cdot Vert_2$.

So given $Vert x Vert =1$, $$Vert AB x Vert le Vert A Vert Vert Bx Vert,$$ the equality holds when $Bx$ hit on the direction of first right singular vector of $A$.

Then
$$Vert Bx Vert le Vert B Vert Vert x Vert$$ the equality holds when $x$ hit the first right singular vector of $B$. However, now the $Bx$ aligned with first left singular vector of $B$ and it must match the first right singular vector of $A$.

Note that $$Vert A B xVert le Vert A Vert Vert BVert Vert x Vert$$

When both equality conditions are holds, that is to say,

the largest left singular vector of $B$ is parallel to the largest right singular vector of $A$

we have
$$
Vert A BVert = sup Vert AB xVert/Vert x Vert = Vert A Vert Vert B Vert
$$