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A matrix A € M₂(C) is Hermitian if A = A, where At is the transpose matrix, and A is the conjugate matrix (if A= (a) then A := (j)). Show that a matrix A € M₂(C) is Hermitian if and only if A is of (1) for some x, y e R and z e C. (2) Let U denote the set of all Hermitian matrices in M₂ (C). (a) Prove that U is NOT a complex vector subspace of M₂(C). (b) Prove that U is a real vector subspace of M₂ (C); here, we view M₂ (C) as a real vector space, under usual matrix addition and real scalar multiplication. the form

A matrix A € M₂(C) is Hermitian if A = A, where At is the transpose matrix, and A is the conjugate matrix (if A= (a) then A := (j)). Show that a matrix A € M₂(C) is Hermitian if and only if A is of (1) for some x, y e R and z e C. (2) Let U denote the set of all Hermitian matrices in M₂ (C). (a) Prove that U is NOT a complex vector subspace of M₂(C). (b) Prove that U is a real vector subspace of M₂ (C); here, we view M₂ (C) as a real vector space, under usual matrix addition and real scalar multiplication. the form

Advanced Engineering Mathematics
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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A matrix A € M₂(C) is Hermitian if A¹ = A, where At is the transpose matrix, and A is the conjugate matrix (if A = (aij) then Ā:= (ij)). Show that a matrix A € M₂(C) is Hermitian if and only if A is of (1) x for some x, y R and z € C. (2) Let U denote the set of all Hermitian matrices in M₂(C). (a) Prove that U is NOT a complex vector subspace of M₂(C). (b) Prove that U is a real vector subspace of M₂ (C); here, we view M₂(C) as a real vector space, under usual matrix addition and real scalar multiplication. (c) Find a basis of U (viewed as a real vector space). the form
Transcribed Image Text:A matrix A € M₂(C) is Hermitian if A¹ = A, where At is the transpose matrix, and A is the conjugate matrix (if A = (aij) then Ā:= (ij)). Show that a matrix A € M₂(C) is Hermitian if and only if A is of (1) x for some x, y R and z € C. (2) Let U denote the set of all Hermitian matrices in M₂(C). (a) Prove that U is NOT a complex vector subspace of M₂(C). (b) Prove that U is a real vector subspace of M₂ (C); here, we view M₂(C) as a real vector space, under usual matrix addition and real scalar multiplication. (c) Find a basis of U (viewed as a real vector space). the form
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