User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Dilation

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Dilation[edit | edit source]

Matrix inequalities can be dilated in order to obtain a larger matrix inequality. This can be a useful technique to separate design variables in a BMI (bi-linear matrix inequality), as the dilation often introduces additional design variables.

A common technique of LMI dilation involves using the projection lemma in reverse, or the "reciprocal projection lemma." For instance, consider the matrix inequality



where , , with This can be rewritten as

(1)


Then since


which is equivalent to


(2)


These expanded inequalities (1) and (2) are now in the form of the strict projection lemma, meaning they are equivalent to


(3)


where and By choosing


we can now rewrite the inequality (3) as


which is the new dilated inequality.

Examples[edit | edit source]

Some useful examples of dilated matrix inequalities are presented here.


Example 1

Consider matrices where and The following matrix inequalities are equivalent:




Example 2

Consider matrices and where The matrix inequality



implies the inequality



Example 3

Consider matrices and where The matrix inequality



implies the inequality


Related Pages[edit | edit source]

External Links[edit | edit source]