# If A, B Hermitian, then <v|AB|v>=<v|BA|v>*. Why?

In summary, the conversation discusses the relationship between Hermitian operators A and B, a vector v, and the inner product notation. It is proven that if <v|AB|v>=x+iy, then <v|BA|v> = x-iy by using the definition of inner product and the properties of Hermitian operators and adjoint operators.
Gold Member
This is certainly an elementary question, so I would be all the more grateful for the answer. Given: A and B are two Hermitian operators and v is a vector in C. If <v|AB|v>=x+iy (for x and y real), then <v|BA|v> = x-iy.
Why?

Bra-ket notation can be confusing, so I will translate to inner product notation. If I denote the inner product of two arbitrary vectors x and y by (x,y), and denote the ket |v> by v, then <v|AB|v> means (v,ABv). The definition of "inner product" says that ##(x,y)^*=(y,x)## for all x,y. This implies that ##(v,BAv)^*=(BAv,v)##. Now you can use that A and B are hermitian, and the definition of the adjoint operator to evaluate the right-hand side.
$$(v,BAv)^*=(BAv,v)=(B^*A^*v,v)=(A^*v,Bv)=(v,ABv).$$

1 person
Many thanks, Fredrik. That clears that step up.

## 1. What does it mean for a matrix to be Hermitian?

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, the matrix is equal to its own conjugate when flipped along the diagonal. This means that all of the elements on the diagonal are real numbers, and the elements above the diagonal are the complex conjugates of the elements below the diagonal.

## 2. Why is it important to know if A and B are Hermitian in the equation =?

Knowing that A and B are Hermitian is important because it allows us to simplify the equation to =*. This simplification is useful in many areas of mathematics and physics, including quantum mechanics and linear algebra.

## 3. Can this equation be applied to non-Hermitian matrices?

No, this equation only holds true for Hermitian matrices. If A and B are not Hermitian, then the equation =* does not hold.

## 4. How does this equation relate to the commutative property of matrix multiplication?

This equation is a special case of the commutative property of matrix multiplication. The commutative property states that the order of multiplication does not matter when multiplying matrices. In this case, the equation shows that the order of multiplication does matter for non-Hermitian matrices, but for Hermitian matrices, it is equivalent to the commutative property.

## 5. In what situations would this equation be useful?

This equation is useful in many areas of mathematics and physics, such as quantum mechanics, linear algebra, and matrix theory. It can be used to simplify calculations and prove theorems involving Hermitian matrices. It is also useful in understanding the properties of quantum mechanical systems, where Hermitian operators play a crucial role.

Replies
2
Views
1K
Replies
3
Views
2K
Replies
5
Views
2K
Replies
21
Views
1K
Replies
10
Views
837
Replies
3
Views
1K
Replies
5
Views
1K
Replies
1
Views
734
Replies
2
Views
2K
Replies
2
Views
2K