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By Nathan Jacobson

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X ) = BX ; XD where the right hand-side is in terms of matrix addition and multiplication. Clearly is a linear mapping since ( X1 + X2 ) = B ( X1 + X2 ) ; ( X1 + X2 )D = (BX1 ; X1 D) + (BX2 ; X2 D) = (X1 ) + (X2 ): If we now consider the identi cation between the matrix space C k l and the product space C kl , then can be thought of as a map from C kl to C kl , and can accordingly be represented by a complex matrix which is kl kl. We now do an explicit 2 2 example for illustration. Suppose k = l = 2 and that B = 13 24 and D = 50 00 : We would like to nd a matrix representation for .

H n is a linear map and the LMI can be written compactly as F (Z ) < ;T: With these examples and de nition in hand, we will easily be able to recognize an LMI. Here we have formulated LMIs in terms of the Hermitian matrices, which is the most general situation for our later analysis. In some problems LMIs are written over the space of symmetric matrices Sn, and this is the usual form employed for computation. 4. Linear Matrix Inequalities 49 to the following discussion, and furthermore in the exercises we will see that the Hermitian form can always be converted to the symmetric form.

Furthermore if A is a real matrix, then any non real eigenvalues must appear in conjugate pairs. Also, a matrix has the eigenvalue zero if and only if it is singular. Associated with every eigenvalue k is the subspace Ek = ker(A ; k I ) every nonzero element in Ek is an eigenvector corresponding to the eigenvalue k . Now suppose that a set of eigenvectors satis es spanfx1 : : : xn g = C n : xn , and from Then we can de ne the invertible matrix X = x1 the matrix product we nd Axn = 1 x1 AX = Ax1 n xn = X where is the diagonal matrix 2 3 0 1 7 ...

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