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(i) Let A be an n x n matrix over C and II be an m x m projection matrix. Let z E C. Calculate exp(z(A > II)). (ii) Let A₁, A2 be n x n matrices over C. Let II₁, I2 be m x m projection matrices with II₁I₂ = On. Calculate exp(z(A₁II₁ + A₂ II₂)). (iii) Use the result from (ii) to find the unitary matrix U(t) = exp(-iĤt/ħ) where Ĥ = ħw(A₁II₁ + A₂II₂) and we assume that A₁ and A2 are hermitian matrices.

(i) Let A be an n x n matrix over C and II be an m x m projection matrix. Let z E C. Calculate exp(z(A > II)). (ii) Let A₁, A2 be n x n matrices over C. Let II₁, I2 be m x m projection matrices with II₁I₂ = On. Calculate exp(z(A₁II₁ + A₂ II₂)). (iii) Use the result from (ii) to find the unitary matrix U(t) = exp(-iĤt/ħ) where Ĥ = ħw(A₁II₁ + A₂II₂) and we assume that A₁ and A2 are hermitian matrices.

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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(i) Let A be an n × n matrix over C and II be an m × m projection matrix. Let z E C. Calculate exp(z(A > II)). (ii) Let A₁, A₂ be n × n matrices over C. Let II₁, II₂ be m x m projection matrices with II₁II₂ = On. Calculate exp(z(A₁II₁ + A₂ > II₂)). (iii) Use the result from (ii) to find the unitary matrix U(t) = exp(-iĤt/ħ) where Ĥ = ħw(A₁ ⓇII₁ + A₂ ❀ II₂) and we assume that A₁ and A₂ are hermitian matrices.
Transcribed Image Text:(i) Let A be an n × n matrix over C and II be an m × m projection matrix. Let z E C. Calculate exp(z(A > II)). (ii) Let A₁, A₂ be n × n matrices over C. Let II₁, II₂ be m x m projection matrices with II₁II₂ = On. Calculate exp(z(A₁II₁ + A₂ > II₂)). (iii) Use the result from (ii) to find the unitary matrix U(t) = exp(-iĤt/ħ) where Ĥ = ħw(A₁ ⓇII₁ + A₂ ❀ II₂) and we assume that A₁ and A₂ are hermitian matrices.
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