Let A e C™×n; that is, A corresponds to a linear operator C" → C*. Define the Hermitian transpose A* of A via the property that (Ax) · y = x : for all x, y E C" (Ac) (recall that here we need to use the inner product on C", which is the complex dot product defined by x · y := x' y = E÷1 xiYi). i=1 (i) Prove that the property Ac uniquely defines the matrix A*, by showing that the matrix entries are given by (A*)ij = āji. Hint: use x = e¡, y = e;, where e; is the jth unit basis vector with (e;)k = 8jk-]

Let A e C™×n; that is, A corresponds to a linear operator C" → C. Define the Hermitian transpose A of A via the property that (Ax) · y = x : for all x, y E C" (Ac) (recall that here we need to use the inner product on C", which is the complex dot product defined by x · y := x' y = E÷1 xiYi). i=1 (i) Prove that the property Ac uniquely defines the matrix A, by showing that the matrix entries are given by (A)ij = āji. Hint: use x = e¡, y = e;, where e; is the jth unit basis vector with (e;)k = 8jk-]

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let A e C"Xn; that is, A corresponds to a linear operator C" → C".
Define the Hermitian transpose A* of A via the property that
(Ax) · y = x ·
(A*y)
for all x, y E C"
(Ac)
(recall that here we need to use the inner product on C", which is the complex dot
product defined by x· y := x' y = E-1 xiYi).
.T
i=1
(i) Prove that the property Ac uniquely defines the matrix A*, by showing that the
matrix entries are given by (A*)ij
a ji.
[Hint: use x = e;, y = e;, where e; is the jth unit basis vector with (e;)k = 8jk•]

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