Let A E Cnxn; that is, A corresponds to a linear operator C" → C".Define the Hermitian transpose A* of A via the property that(Ах) у — х: (А'у)for all x, у Е С"(Ac)(recall that here we need to use the inner product on C", which is the complex dotproduct defined by x y := x"y = E-1 c;Y5).(i) Prove that the property Ac uniquely defines the matrix A*, by showing that thematrix entries are given by (A*)ij = aji.[Hint: use x = e¡, y = e;, where e; is the jth unit basis vector with (e;)k = 8jk.]= AT (that is, let AH be the matrix with entries(ii) Going the other way: define A(A“)ij = aji), and show that(Ах) у — х:(A"y)for all x, у € C".This demonstrates that AH = AT satisfies the property Ac above.Together, (a) and (b) show that in the complex case, the matrix with the desired propertyAc– called the Hermitian transpose, or adjoint (typically denoted by A* or A“) – isgiven by the (complex) conjugate transpose.-(iii) Conclude that for complex-valued matrices, the symmetry property A = A*, that is,(Ах) у 3D х: (Ау)for all x, у € С",(Sc)is equivalent to aij = aji.

Part(i): Let, A∈ℂn×n, let Aℂ be the following property, Ax·y=x·A*y ∀x,y∈ℂn Then, A*ij=aji Now,…

et A e C™×n; that is, A corresponds to a linear operator C" → C". efine the Hermitian transpose A* of A via the property that (Ах) у — х (А"у) for all x, у Є С" (Ac. ecall that here we need to use the inner product on C", which is the complex dot roduct defined by x · y - x "y = ΣΗ1 0.1). := X ) Prove that the property Ac uniquely defines the matrix A*, by showing that the natrix entries are given by (A*)ij = aji. Hint: use x = e¿, y = ej, where ej is the jth unit basis vector with (e;)k = 8jk-] i) Going the other way: define AH A")ij = aji), and show that AT (that is, let A" be the matrix with entries %3D

et A e C™×n; that is, A corresponds to a linear operator C" → C". efine the Hermitian transpose A* of A via the property that (Ах) у — х (А"у) for all x, у Є С" (Ac. ecall that here we need to use the inner product on C", which is the complex dot roduct defined by x · y - x "y = ΣΗ1 0.1). := X ) Prove that the property Ac uniquely defines the matrix A, by showing that the natrix entries are given by (A)ij = aji. Hint: use x = e¿, y = ej, where ej is the jth unit basis vector with (e;)k = 8jk-] i) Going the other way: define AH A")ij = aji), and show that AT (that is, let A" be the matrix with entries %3D

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 Cnxn; that is, A corresponds to a linear operator C" → C".
Define the Hermitian transpose A* of A via the property that
(Ах) у — х: (А'у)
for all x, у Е С"
(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 c;Y5).
(i) Prove that the property Ac uniquely defines the matrix A*, by showing that the
matrix entries are given by (A*)ij = aji.
[Hint: use x = e¡, y = e;, where e; is the jth unit basis vector with (e;)k = 8jk.]
= AT (that is, let AH be the matrix with entries
(ii) Going the other way: define A
(A“)ij = aji), and show that
(Ах) у — х:
(A"y)
for all x, у € C".
This demonstrates that AH = AT satisfies the property Ac above.
Together, (a) and (b) show that in the complex case, the matrix with the desired property
Ac– called the Hermitian transpose, or adjoint (typically denoted by A* or A“) – is
given by the (complex) conjugate transpose.
-
(iii) Conclude that for complex-valued matrices, the symmetry property A = A*, that is,
(Ах) у 3D х: (Ау)
for all x, у € С",
(Sc)
is equivalent to aij = aji.

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