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# lmiinfo

Information about variables and term content of LMIs

## Syntax

```lmiinfo
```

## Description

lmiinfo provides qualitative information about the system of LMIs lmisys. This includes the type and structure of the matrix variables, the number of diagonal blocks in the inner factors, and the term content of each block.

lmiinfo is an interactive facility where the user seeks specific pieces of information. General LMIs are displayed as

```N' * L(x) * N < M' * R(x) * M
```

where N,M denote the outer factors and L,R the left and right inner factors. If the outer factors are missing, the LMI is simply written as

```L(x) < R(x)
```

If its right side is zero, it is displayed as

```N' * L(x) * N < 0
```

Information on the block structure and term content of L(x) and R(x) is also available. The term content of a block is symbolically displayed as

```C1 + A1*X2*B1 + B1'*X2*A1' + a2*X1 + x3*Q1
```

with the following conventions:

• X1, X2, x3 denote the problem variables. Upper-case X indicates matrix variables while lower-case x indicates scalar variables. The labels 1,2,3 refer to the first, second, and third matrix variable in the order of declaration.

• Cj refers to constant terms. Special cases are I and –I (I = identity matrix).

• Aj, Bj denote the left and right coefficients of variable terms. Lower-case letters such as a2 indicate a scalar coefficient.

• Qj is used exclusively with scalar variables as in x3*Q1.

The index j in Aj, Bj, Cj, Qj is a dummy label. Hence C1 may appear in several blocks or several LMIs without implying any connection between the corresponding constant terms. Exceptions to this rule are the notations A1*X2*A1' and A1*X2*B1 + B1'*X2'*A1' which indicate symmetric terms and symmetric pairs in diagonal blocks.

## Examples

Consider the LMI

$0\left(\begin{array}{cc}-2X+{A}^{T}YB+{B}^{T}{Y}^{T}A+I& XC\\ {C}^{T}X& -zI\end{array}\right)$

where the matrix variables are X of Type 1, Y of Type 2, and z scalar. If this LMI is described in lmis, information about X and the LMI block structure can be obtained as follows:

```lmiinfo(lmis)

LMI ORACLE
-------

This is a system of 1 LMI with 3 variable matrices

Do you want information on
(v) matrix variables     (l) LMIs     (q) quit

?> v

Which variable matrix (enter its index k between 1 and 3) ? 1
X1 is a 2x2 symmetric block diagonal matrix
its (1,1)-block is a full block of size 2

-------

This is a system of 1 LMI with 3 variable matrices
Do you want information on
(v) matrix variables     (l) LMIs     (q) quit

?> l

Which LMI (enter its number k between 1 and 1) ? 1

This LMI is of the form
0 < R(x)
where the inner factor(s) has 2 diagonal block(s)

Do you want info on the right inner factor ?

(w) whole factor     (b) only one block

?> w

Info about the right inner factor

block (1,1) : I + a1*X1 + A2*X2*B2 + B2'*X2'*A2'

block (2,1) : A3*X1

block (2,2) : x3*A4

(w) whole factor     (b) only one block

-------

This is a system of 1 LMI with 3 variable matrices

Do you want information on
(v) matrix variables     (l) LMIs     (q) quit

?> q

It has been a pleasure serving you!
```

Note that the prompt symbol is ?> and that answers are either indices or letters. All blocks can be displayed at once with option (w), or you can prompt for specific blocks with option (b).