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The eigenvalues of a Hermitian matrix are always real 📂Matrix Algebra

The eigenvalues of a Hermitian matrix are always real

Theorem

Let AA be a Hermitian matrix of size n×nn \times n. Then, the eigenvalues of AA are all real.

Explanation

In general, there is no guarantee that the eigenvalues of a matrix are real, but for Hermitian matrices, this can be verified through proof.

It may not be intuitively obvious, but the proof itself is relatively simple, and it is quite useful as a fact. It yields various good results when combined with concepts like positive-definiteness, so it is well worth knowing.

Proof

Let’s say the eigenvalue of AA is λ\lambda and the corresponding eigenvector is x\mathbf{x}. Then, the eigenvalue equation is as follows.

Ax=λx A \mathbf{x} = \lambda \mathbf{x}

Multiplying both sides by x∗\mathbf{x}^{\ast} yields:

x∗Ax=λx∗x \begin{equation} \mathbf{x}^{\ast} A \mathbf{x} = \lambda \mathbf{x}^{\ast} \mathbf{x} \end{equation}

Taking the conjugate transpose ∗^{ \ast } on both sides, by the properties of conjugate transpose, we get:

(x∗Ax)∗=(λx∗x)∗  ⟹  x∗A∗x=λ‾x∗x \begin{align*} \left( \mathbf{x}^{\ast} A \mathbf{x} \right) ^{\ast} = & \left( \lambda \mathbf{x}^{\ast} \mathbf{x} \right) ^{\ast} \\ \implies \mathbf{x}^{\ast} A^{ \ast } \mathbf{x} = & \overline{\lambda} \mathbf{x}^{ \ast } \mathbf{x} \end{align*}

Since AA is a Hermitian matrix, we have A=A∗A=A^{\ast}, and the equation can be rewritten as:

x∗Ax=λ‾x∗x \begin{equation} \mathbf{x}^{\ast} A \mathbf{x} = \overline{\lambda} \mathbf{x}^{ \ast } \mathbf{x} \end{equation}

By (1)(1) and (2)(2), the following equation holds:

λx∗x=x∗Ax=x∗A∗x=λ‾x∗x \lambda \mathbf{x}^{ \ast } \mathbf{x} = \mathbf{x}^{ \ast } A \mathbf{x} = \mathbf{x}^{ \ast } A^{ \ast } \mathbf{x} = \overline{\lambda} \mathbf{x}^{ \ast } \mathbf{x}

Therefore, (λ−λ‾)x∗x=0 ( \lambda - \overline{\lambda} ) \mathbf{x}^{ \ast } \mathbf{x} = 0

However, since x\mathbf{x} is an eigenvector, it is not 0\mathbf{0}. Therefore, x∗x≠0\mathbf{x}^{\ast} \mathbf{x} \ne \mathbf{0} must be true. Hence,

λ=λ‾ \lambda = \overline{\lambda}

This implies that λ\lambda is real.

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