3. Let G be a 4 by 4 complex matrix: G= 1 - -i (a) (2...

  

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3. Let G be a 4 by 4 complex matrix: G= 1 - -i (a) (2 points) This matrix has two eigenvalues A=2, and one eigenvalue A=-2. Find the fourth eigenvalue. (b) (2 points) Find a real eigenvector and show it is indeed an eigenvector. (c) (2 points) Is G Hermitian? Why or why not? (G is Hermitian if GT = G, the bar indicates complex conjugate.) == (d) (2 points) Give an example of a real non-diagonal matrix X for which GHXG is Hermitian. 3. Let G be a 4 by 4 complex matrix: G= 1 - -i (a) (2 points) This matrix has two eigenvalues A=2, and one eigenvalue A=-2. Find the fourth eigenvalue. (b) (2 points) Find a real eigenvector and show it is indeed an eigenvector. (c) (2 points) Is G Hermitian? Why or why not? (G is Hermitian if GT = G, the bar indicates complex conjugate.) == (d) (2 points) Give an example of a real non-diagonal matrix X for which GHXG is Hermitian.