Exponential of hermitian matrix

In summary: If you expand this, you will get a sum of terms, each of which is ##a_i^m## times the original eigenvector ##\nu_i##In summary, The conversation discusses the properties of a Hermitian matrix and its corresponding matrix U defined by the Taylor expansion of the exponential. The first part of the conversation focuses on showing that the eigenvectors of A are also eigenvectors of U, with the eigenvalues of U being given by the exponential of the negative imaginary component of the eigenvalues of A. The second part discusses the importance of this example in quantum mechanics and how U is unitary. It is suggested to input the eigenvalue equation for A into the Taylor expansion definition of U and to use
  • #1
ZCOR
3
0

Homework Statement



Let A be a Hermitian matrix and consider the matrix U = exp[-iA] defined by thr Taylor expansion of the exponential.

a) Show that the eigenvectors of A are eigenvectors of U. If the eigenvalues of A are a subscript(i) for i=1,...N, show that the eigenvalues of U are exp[-i*a subscript(i)].

b) Show that U is unitary.

This example is important in quantum mechanics when A=Ht/h, where H is the Hamiltonian operator, t is time, and h is Plank's constant. Then U evolves the wavefunction over a time t.

Homework Equations


I have uploaded what I have come up with so far, but not sure where to go with it

The Attempt at a Solution


[/B]
See uploaded file
 

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  • #2
You have ##A\nu_i = a_i \nu_i## where ##a_i## is the eigenvalue corresponding to an eigenvector ##\nu_i##. Have you tried inputting this eigenvalue equation into the eigenvalue equation for ##U## using its Taylor expansion definition?
ZCOR said:
Show that U is unitary.
Do you know the property of the determinant of a unitary matrix?
 
  • #3
Ok, will see if I can make that work. Thanks!
 
  • #4
I'm trying to figure this out, but not sure how to input that equation into the Taylor expansion
 
  • #5
Try applying ##U## in its Taylor form to an eigenvector of ##A##
$$
(I + iA - \frac{1}{2}A^2 - i\frac{1}{6} A^3 + \ldots) \nu_i .
$$
 

Related to Exponential of hermitian matrix

1. What is the definition of the exponential of a Hermitian matrix?

The exponential of a Hermitian matrix is a special function that takes a Hermitian matrix as its input and produces another matrix as its output. It is defined as the infinite sum of powers of the matrix, with the coefficients determined by the matrix's eigenvalues and eigenvectors. This function is commonly denoted as eA, where A is the Hermitian matrix.

2. How is the exponential of a Hermitian matrix calculated?

The exponential of a Hermitian matrix can be calculated using the matrix's eigenvalues and eigenvectors. The formula for calculating the exponential is eA = PeDP-1, where P is the matrix of eigenvectors and D is the diagonal matrix with the eigenvalues on the diagonal. This formula is known as the spectral decomposition of a Hermitian matrix.

3. What are the properties of the exponential of a Hermitian matrix?

The exponential of a Hermitian matrix has several important properties, including:

  • The exponential of a Hermitian matrix is always a unitary matrix.
  • The exponential of a Hermitian matrix is always positive definite.
  • The exponential of a Hermitian matrix is always diagonalizable.
  • The exponential of a Hermitian matrix is always a real-valued matrix.

4. Why is the exponential of a Hermitian matrix important?

The exponential of a Hermitian matrix is important in many areas of mathematics and physics. It is used in quantum mechanics to describe time evolution of quantum systems. It is also used in differential equations to solve problems involving linear transformations. In addition, the exponential of a Hermitian matrix has applications in signal processing, control theory, and data science.

5. Are there any limitations to the exponential of a Hermitian matrix?

Yes, there are limitations to the exponential of a Hermitian matrix. One limitation is that the exponential function is only defined for square matrices. Additionally, the eigenvalues of the matrix must be distinct in order for the spectral decomposition formula to hold. Finally, the exponential of a Hermitian matrix may not always be easy to calculate, as it involves finding the eigenvalues and eigenvectors of the matrix.

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