# Can eigenvalues be imaginary?

## Can eigenvalues be imaginary?

The main difference between imaginary and real eigenvalues is that imaginary eigenvalues are imaginary, whereas real eigenvalues are real.

## Which one of the following matrices is skew-Hermitian?

When the conjugate transpose of a complex square matrix is equal to the negative of itself, then this matrix is called as skew hermitian matrix. If P is a complex square matrix and if it satisfies Pθ = -P then such matrix is termed as skew hermitian.

Is skew-Hermitian matrix invertible?

The result implies that every odd degree skew-symmetric matrix is not invertible, or equivalently singular.

How do you find the eigenvalues of a Hermitian matrix?

Eigenvectors of a Hermitian matrix corresponding to distinct eigenvalues are mutually orthogonal. u∗Au = u∗(λu) = λ(u∗u) = λu2. Since u∗Au is real and u is a nonzero real number, it follows that λ is real. λu∗v = (λu)∗v = (Au)∗v = u∗(Av) = u∗µv = µ(u∗v).

### What are the eigenvalues of skew Hermitian matrix?

The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero. Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.

### What is the difference between skew-Hermitian matrix and real numbers?

matrices), whereas real numbers correspond to self-adjoint operators. The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero).

How to prove that eigenvalues are purely imaginary?

Then, A † = V Λ ∗ V †, and therefore A † = − A ⟺ Λ ∗ = − Λ, which implies that the eigenvalues are purely imaginary. Show activity on this post. And λ is purely imaginray.

What is a skew-symmetric matrix?

A skew-symmetric matrix is a matrix whose transpose is equal to the negative of the matrix. In the same way, a skew Hermitian matrix is a matrix whose conjugate transpose is equal to the negative of the matrix. Let us learn how to identify a skew Hermitian matrix and its properties along with more examples.