aHermitian Matrices

1. Introduction 2. Conjugate Transpose 3. Hermitian Matrices 4. 2×2 Hermitian Matrices 5. Observations and Properties 6. Real Symmetric Matrices

1. Introduction We now turn our attention to matrices. We are familiar with real matrices. A complex matrix will have complex entries. A simple example follows: A=a

i	2+i	-1+3i
0	-4	1-5i

A is a 2×3 complex matrix. A word about notations: • Matrices will be represented using bold capital letters.• Vectors will be represented using bold small letters. • Scalars will be represented using plain small letters. Most of the operations on real matrices carry over to their complex counterparts. Addition, matrix multiplication, transpose are all similar. One important addition to the list of operations is the conjugate transpose. 2. Conjugate Transpose What is true for a vector can be extended to the case of any general m×n matrix. Given a complex matrix A, its conjugate transpose is denoted by AH or A*. A simple example follows: A=a

1	2i
3+i	5
1-i	0

⟹A*=a

1	3-i	1+i
-2i	5	0

A is a 3×2 matrix and A* is a 2×3 matrix. As noted earlier, the order of operations doesn't matter. We could do the transpose first and then the conjugate or vice-versa. Now, let us quickly state two useful properties of the conjugate transpose: 1. (A*)*=A 2. (AB)*=B*A* The real counterparts of these two properties are (AT)T=A and (AB)T=BTAT. 3. Hermitian Matrices Recall the idea of a symmetric real matrix. A matrix A is symmetric if AT=A. When it comes to complex matrices, the conjugate transpose gains the upper hand over the simple transpose. The complex analogue of a real symmetric matrix is a Hermitian matrix:

A matrix A is a Hermitian matrix if A*=A.

An example of a Hermitian matrix is as follows: A=a

1	2+4i
2-4i	3

Then, A*=a

1	2+4i
2-4i	3

We see that A*=A, therefore A is a Hermitian matrix. Note that a Hermitian matrix has to be a square matrix. An alternative way of defining a Hermitian matrix is using the individual elements. If Aij denotes the jth element in the ith row of the matrix, then for a Hermitian matrix:Aij=⏨⏨Aji 4. 2×2 Hermitian Matrices Let us now try to find out a general form for all 2×2 Hermitian matrices. Let A be some arbitrary 2×2 complex matrix, with complex numbers a,b,c,d as its entries: A=a

a	b
c	d

If A has to be Hermitian, then A*=A. This leads us to: a

⏨a	⏨c
⏨b	⏨d

=a

a	b
c	d

Two matrices are equal if and only if their corresponding elements are equal. Therefore, we have the following equalities: a

⏨a	=a
⏨d	=d
⏨b	=c
⏨c	=b

A complex number is equal to its conjugate if and only if it is a real number, which is same as saying that its imaginary part is 0. This means that a and d are real numbers. b and c are conjugates of each other. We now have all the ingredients to write the general form of any 2×2 Hermitian matrix. With a slight abuse of notation, for real numbers a,b,c,d, A as given below is a Hermitian matrix: A=a

c	a+ib
a-ib	d

In fact, every 2×2 Hermitian matrix, is associated with a unique tuple (a,b,c,d) of real numbers. Note that if b=0, then A turns into a real symmetric matrix. 5. Observations and Properties 1. All real symmetric matrices are Hermitian. If A is symmetric and real, then AT=A and ⏨A=A. From this, it follows that A*=A.1. 2. The diagonal entries of a Hermitian matrix are real. We saw this in the case of 2×2 Hermitian matrices. For the more general case of an n×n Hermitian matrix, we can see that Aii=⏨⏨Aii. It follows that the diagonal elements have to be real.2. 3. All the eigenvalues of a Hermitian matrix are real. Let A be Hermitian. Let (𝜆,v) be an eigenvalue-eigenvector pair or eigenpair of A, then we have: a

𝜆v*v	=v*(𝜆v)
	=v*(Av)
	=v*Av
	=v*A*v
	=(v*A*)v
	=(Av)*v
	=(𝜆v)*v
	=⏨𝜆v*v

From this, we see that: a

v*v(⏨𝜆-𝜆)	=0
⏨𝜆	=𝜆

We can now conclude that 𝜆∈R. We have used the fact that v≠0 as it is an eigenvector. Hence, v*v≠0. 4. Eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal. That is, if (𝜆1,v1) and (𝜆2,v2) are two eigenpairs of a Hermitian matrix A with 𝜆1≠𝜆2, then v1⟂v2. Let us prove this: a

Av1	=𝜆1v1
v*2Av1	=𝜆1v*2v1
v*2A*v1	=𝜆1v*2v1
(Av2)*v1	=𝜆1v*2v1
(𝜆2v2)*v1	=𝜆1v*2v1
⏨𝜆2v*2v1	=𝜆1v*2v1
𝜆2v*2v1	=𝜆1v*2v1
(𝜆2-𝜆1)v*2v1	=0
⟹v*2v1	=0

We can see that v*2v1=0 and hence v1⟂v2. 5. Given a Hermitian matrix A, for any vector x, the quantity x*Ax is a real number. To prove this, recall that a complex number z is a real number if and only if ⏨z=z. Also recall that if x,y are two vectors, then ⏨⏨x*y=y*x (conjugate symmetry). We shall use these two as a part of the proof: a

⏨⏨⏨x*Ax	=⏨⏨⏨x*A*x
	=⏨⏨⏨⏨x*(A*x)
	=(A*x)*x
	=x*(A*)*x
	=x*Ax

Since the conjugate of x*Ax is equal to it, x*Ax has to be a real number. 6. Real Symmetric Matrices As stated earlier in the section on properties of Hermitian matrices, every real symmetric matrix is Hermitian. So, all that we have discussed regarding Hermitian matrices holds good for real symmetric matrices. Since we are dealing with real matrices, the conjugate transpose reduces to the just the transpose. Likewise, the complex inner product reduces to the simple dot product. We will briefly restate two important properties in the special case of real symmetric matrices: 1. All the eigenvalues of a real symmetric matrix are real.2. The eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are orthogonal.