Variation of linear matrix inequality

Question

When reading "Convex optimization, S. Boyd" p.76, Example 3.4, it says

The last condition is a linear matrix inequality (LMI) in $(x,Y,t)$. Therefore, epi($f$) is convex.

I am confused about why this form is called LMI? (I know LMI, but am confused in this special form.)

$A(x) = x_1A + ... + x_nA \leq B$ is called a LMI in $x$

Answer 1

Linear Matrix Inequalities should have been called Affine Matrix Inequalities but it got stuck with the first. The second form that is confusing you is the base form. The screenshot is in the more natural block variable form. Assume you have a 2x2 LMI given as

$$ \begin{bmatrix} a&b\\b&1 \end{bmatrix} \succ 0 $$

Then from this block form to the basis is

$$ a\begin{bmatrix} 1&0\\0&0 \end{bmatrix} + b\begin{bmatrix} 0&1\\1&0 \end{bmatrix} \succ \begin{bmatrix} 0&0\\0&-1 \end{bmatrix}$$

The left hand side is all linear in the unknowns hence the name ignoring the $B$.

Since it is super labourous to write the basis form all the time, you write as the first form and you are done. The parser Yalmip for example is mainly responsible for converting the first to the second form (among other issues).