# Hermitian Inverse: Exploring Eigenvalues of ##H^{-1}##

• I
• ergospherical
In summary, the conversation discusses a Hermitian matrix ##H## with eigenvectors ##\mathbf{e}_i## and all eigenvalues negative. It is claimed that ##G = \int_{0}^{\infty} e^{tH} dt## satisfies ##G = H^{-1}##. However, when looking at ##G\mathbf{e}_i##, it is found to be equal to ##-1/\lambda_i \mathbf{e}_i##, which contradicts the eigenvalues of ##H^{-1}## being ##1/\lambda_i##. It is then discovered that this only applies to specific cases, such as when ##H = -I##.
ergospherical
##H## is an ##n\times n## Hermitian matrix with eigenvectors ##\mathbf{e}_i## and all eigenvalues negative. It's claimed that ##G = \int_{0}^{\infty} e^{tH} dt## is such that ##G = H^{-1}##. I was looking at\begin{align*}
G\mathbf{e}_i &= \int_0^{\infty} \sum_{n=1}^{\infty} \frac{t^n}{n!} H^n \mathbf{e}_i dt = \mathbf{e}_i\int_0^{\infty} \sum_{n=1}^{\infty} \frac{t^n \lambda_i^n}{n!} dt = \mathbf{e}_i \int_0^{\infty} e^{\lambda_i t} dt = - \frac{1}{\lambda_i} \mathbf{e}_i
\end{align*}which is weird, because ##\mathbf{e}_i = H^{-1} H\mathbf{e}_i = \lambda_i H^{-1} \mathbf{e}_i## so ##H^{-1}## should have eigenvalues ##1/\lambda_i##?

If you try ##H = -I## you'll see you are correct. ##G = -H^{-1}##.

ergospherical
Thanks, I'm surprised. It's part of an old exam question!

## What is a Hermitian Inverse?

A Hermitian inverse, also known as a conjugate transpose, is the inverse of a Hermitian matrix. It is obtained by taking the transpose of the matrix and then taking the complex conjugate of each element.

## What is the significance of exploring eigenvalues of ##H^{-1}##?

Exploring the eigenvalues of ##H^{-1}## can provide insight into the properties and behavior of the Hermitian inverse. It can also be used to solve systems of linear equations and analyze the stability of a matrix.

## How are eigenvalues of ##H^{-1}## calculated?

The eigenvalues of ##H^{-1}## can be calculated by taking the inverse of the eigenvalues of the original Hermitian matrix. This can be done using various methods such as diagonalization or the characteristic polynomial.

## What is the relationship between eigenvalues of ##H^{-1}## and the original Hermitian matrix?

The eigenvalues of ##H^{-1}## and the original Hermitian matrix are reciprocals of each other. This means that if an eigenvalue of the original matrix is 0, the corresponding eigenvalue of the Hermitian inverse will be undefined.

## What are some applications of exploring eigenvalues of ##H^{-1}##?

Exploring eigenvalues of ##H^{-1}## has many applications in fields such as physics, engineering, and computer science. It is used in solving systems of linear equations, analyzing the stability of a system, and in quantum mechanics for understanding the behavior of particles.

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