# Documentation

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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 (o) other LMI (t) back to top level ?> 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 (o) other LMI (t) back to top level ------- 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)`.

## Tips

`lmiinfo` does not provide access to the numerical value of LMI coefficients.