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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