aUnitary Matrices

1. Introduction 2. Properties 3. Example 4. Orthogonal Matrices

1. Introduction

A matrix is unitary if it is square and has orthonormal columns.

Unitary matrices are usually denoted by U. If u1,⋯,un are n orthonormal vectors that belong to Cn, then we can construct the following unitary matrix: U=a

|		|
u1	⋯	un
|		|

Let us work with the case of a 2×2 unitary matrix. We have u1,u2∈C2 with u*1u2=0 and u*1u1=u*2u2=1. U=a

|	|
u1	u2
|	|

Notice the following: a

U*U	=a ― u*1 ― ― u*2 ― a | | u1 u2 | |
	=a u*1u1 u*1u2 u*2u1 u*2u2
	=a 1 0 0 1
	=I

We can also show that U is invertible. To see this, note that since u1 and u2 are orthogonal, they are also linearly independent (refer note on Hermitian matrices-1). This means that the columns of U are linearly independent, hence U is invertible. So, we see that U*U=UU*=I and U-1=U*. Thus, another way of defining unitary matrix is as follows:

A unitary matrix U is a n×n matrix that satisfies UU*=U*U=I.

Though we have only seen this for the case of a 2×2 matrix, the definition holds good for any general n×n matrix. Now, for an example: U=1

5a

3	4i
4i	3

Then:a

U*U	=1 25a 3 -4i -4i 3 a 3 4i 4i 3
	=1 25a 25 0 0 25
	=I

Notice that the columns of U are 1

5a

3

4i

and 1

5a

4i

3

. We see that they are unit norm and orthogonal to each other. We will just look at the orthogonal part here: a

u*1u2	=1 25a 3 -4i a 4i 3
	=1 25(12i-12i)
	=0

2. Properties 1. Unitary matrices preserve lengths. If x∈Cn, then its length is ||x||. The claim is that ||Ux|| is the same as ||x||. Let us prove this: a

||Ux||2	=(Ux)*(Ux)
	=x*U*Ux
	=x*(U*U)x
	=x*x
	=||x||2

2. Eigenvalues of a unitary matrix have absolute value of 1. Let U be an unitary matrix and let (𝜆,x) be an eigenpair. From property-(1), we know that unitary matrices preserve lengths: a

||Ux||2	=||x||2
(Ux)*(Ux)	=||x||2
(𝜆x)*(𝜆x)	=||x||2
⏨𝜆𝜆(x*x)	=||x||2
|𝜆|2||x||2	=||x||2
||x||2(|𝜆|2-1)	=0

Since x is an eigenvector, ||x||2≠0. This means that |𝜆|2=1. 3. Eigenvectors corresponding to distinct eigenvalues of a unitary matrix are orthogonal. Let U be a unitary matrix and (𝜆1,v1) and (𝜆2,v2) be two eigenpairs with 𝜆1≠𝜆2. Then: a

v*1v2	=v*1Iv2
	=v*1U*Uv2
	=(Uv1)*(Uv2)
	=(𝜆1v1)*(𝜆2v2)
	=⏨𝜆1𝜆2(v*1v2)
v*1v2(⏨𝜆1𝜆2-1)	=0

Now, we have two possibilities, either v*1v2=0 or ⏨𝜆1𝜆2=1. How do we rule out the second possibility? For a moment, let us assume that ⏨𝜆1𝜆2=1 and see where that takes us: a

⏨𝜆1𝜆2	=1
𝜆1⏨𝜆1𝜆2	=𝜆1
|𝜆1|2𝜆2	=𝜆1
𝜆2	=𝜆1

This is a contradiction as 𝜆1≠𝜆2. Hence, we can conclude that v*1v2=0, which implies that v1⟂v2. 3. Example Let us compute the eigenvalues of the following unitary matrix: U=1

5a

3	4i
4i	3

We have: a

|U-𝜆I|	=0
a 35 -𝜆 4i5 4i5 35 -𝜆	=0
( 35 -𝜆)2-( 4i5 )2	=0
(3-5𝜆)2+16	=0
3-5𝜆	=±4i
𝜆	=3±4i 5

Notice that the eigenvalues are complex numbers with unit absolute value as expected. 4. Orthogonal Matrices An orthogonal matrix is the real analogue of the unitary matrix. So, the columns of an orthogonal matrix are orthonormal and real. Clearly, every orthogonal matrix is unitary. Orthogonal matrices are usually denoted by the letter Q and satisfy the following condition: QTQ=QQT=I The properties discussed in the earlier section carry over here. Since we are working with real matrices, we can replace the complex inner product with the dot product and the conjugate transpose with the transpose. Here is a small example of an orthogonal matrix: Q=1

2a

1	1
-1	1

We shall compute QTQ now: a

QTQ	=1 2a 1 -1 1 1 a 1 1 -1 1
	=1 2a 2 0 0 2
	=I