How to check if a matrix is Hermitian?

Essential Linear algebra lessons for Quantum engineering.

May 14, 2026

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In the last post, I shared with you 5 questions that will help you become great at linear algebra and ace Quantum computing.

Let’s dive into the solution to the second one.

Question 2

Given the following matrix A:

Prove that it is Hermitian
Find its eigenvalues and eigenvectors

Solution 2

2.1: Proving that the matrix is Hermitian

We learned in previous lessons that a Hermitian matrix is a square matrix equal to its conjugate transpose.

The given matrix A is a 2 x 2 square matrix.

Next, we need to find its conjugate transpose, which is a combination of the transpose and complex conjugate operations.

The transpose of A is calculated as:

Next, the complex conjugate is calculated as:

This is the same as A.

This means that A is a Hermitian matrix.

2.2: Finding the eigenvalues and eigenvectors of the matrix

To find the eigenvalues of a square matrix, we need to solve the characteristic equation, defined as follows, where I is the identity matrix.

where

The determinant of this term is calculated as:

Using the quadratic formula on this equation results in:

Hermitian matrices represent observables (physical quantities that can be measured, such as spin, position, momentum, energy, and more) in quantum mechanics.

A core mathematical property of a Hermitian matrix is that all its eigenvalues are real, which we see in this case.

Given a square matrix A of size n x n, an Eigenvector is a non-zero vector v that, when multiplied by A, results in a scalar multiple of itself.

Eigenvectors multiplied by A just scales (stretches or compresses) them without rotation or change in direction.

Let’s next find the eigenvectors using the following equation (derived by rearranging the characteristic equation):