a is a unitary matrix. then eigen value of a are
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Unitary Matrix: If matrix A is called Unitary matrix then it satisfy this condition A.Aθ=Aθ.A=I
where Aθ= Transpose Conjugate of A=(A′)T (first you Conjugate andthen Transpose , you will get Unitary matrix)
Properties of Unitary matrix:
If A is a Unitary matrix thenA−1 is also a Unitary matrix.
If A is a Unitary matrix then Aθ is also a Unitary matrix.
If A&B are Unitary matrices, then A.B is a Unitary matrix.
If A is Unitary matrix then A−1=Aθ
If A is Unitary matrix then it's determinant is of Modulus Unity (always1).
Let A = [
1+i
2
−i+1
2
1+i
2
1−i
2
] is Unitary matrix
Characteristics equation of matrix A is ∣A−λI∣=0
|
1+i−2λ
2
−1+i
2
1+i
2
1−i−2λ
2
|=0
⟹(
1+i−2λ
2
)(
1−i−2λ
2
)−(
1+i
2
)(
i−1
2
)=0
4λ2−4λ+2
4
−
(−2)
4
=0
⟹4λ2−4λ+4=0
⟹4(λ2−λ+1)=0
⟹λ2−λ+1=0
λ=
−(−1)±
√
(−1)2−4(1)(1)
2
⟹λ=
1±
√
1−4
2
⟹λ=
1±
√
−3
2
⟹λ=
1±
√
−1
.
√
3
2
\implies\lambda = \frac{1 + \sqrt{3}.i }{2} , \frac{1 - \sqrt{3}.i }{2}