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In particular, this gives A;1 kAk;1 if A is invertible. ) 3. kUAV k = kAk, and kUAV kF = kAkF , for any appropriately dimensioned unitary matrices U and V . 4. kAB kF kAk kB kF and kAB kF kB k kAkF . Note that although pre-multiplication or post-multiplication of a unitary matrix on a matrix does not change its induced 2-norm and F -norm, it does change its eigenvalues. For example, let A = 11 00 : Then 1 (A) = 1 2 (A) = 0. Now let U= " p12 ; p12 then U is a unitary matrix and # p12 p12 p 2 0 0 0 UA = p with 1 (UA) = 2, 2 (UA) = 0.

Compression is always norm non-increasing and dilation is always norm non-decreasing. Sometimes dilation can be made to be norm preserving. Norm preserving dilations are the focus of this section. 3), we will see that it contains all the essential features of the general problem. 4). 17 8 0, i there is a Y with kY k 1 such that X A X = Y ( 2 I ; A A)1=2 : Proof. X A i X X +A A 2I i X X ( 2 I ; A A): Now suppose X X ( 2 I ; A A) and let h Y := X ( 2 I ; A A)1=2 then X = Y ( 2 I ; A A)1=2 and Y Y Y Y I then X X ( 2 I ; A A): i+ I .

Then there exists a 0 6= v 2 Rn such that v eAt B = 0 for all t 2 0 t1 ]. Therefore, set t = 0, and we have v B = 0: Next, evaluate the i-th derivative of v eAt B = 0 at t = 0 to get v Ai B = 0 i > 0: Hence, we have v B AB A2 B : : : An;1 B = 0 or, in other words, the controllability matrix C does not have full row rank. This is again a contradiction.