Question #46093

a) Check whether the forms 2x

2

+ 3y

2

+ 5z

2

4xz 6yz and 4x

2

+ 3y

2

+ z

2

6xy 2xz

are orthogonally equivalent. (3)

b) Use Gram-Schmidt orthogonalisation process to find an orthonormal basis for the

subspace of C

4

generated by the vectors (1; i; 0; i), ( i; 0; 1; 2) and (0; i; 1; 1). (5)

c) Which of the following matrices are Hermitian and which are Unitary? Justify your

answer. (4)

A =

2

4

1 i 0

i 1 1 i

0 1 + i 2

3

5

; B =

2

4

1

p

2

0

1

p

2

0 1 0

p

2i 0

p

2i

3

5

2

+ 3y

2

+ 5z

2

4xz 6yz and 4x

2

+ 3y

2

+ z

2

6xy 2xz

are orthogonally equivalent. (3)

b) Use Gram-Schmidt orthogonalisation process to find an orthonormal basis for the

subspace of C

4

generated by the vectors (1; i; 0; i), ( i; 0; 1; 2) and (0; i; 1; 1). (5)

c) Which of the following matrices are Hermitian and which are Unitary? Justify your

answer. (4)

A =

2

4

1 i 0

i 1 1 i

0 1 + i 2

3

5

; B =

2

4

1

p

2

0

1

p

2

0 1 0

p

2i 0

p

2i

3

5

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