Real Eigenvalues of Symmetric and Hermitian Matrices

1. The Proof: Why Eigenvalues Must Be Real

Let's dive into the proof. The logic is the same for both symmetric and Hermitian matrices, but we'll start with the real symmetric case and then see how it naturally extends to the complex Hermitian case. The key to the proof is the complex inner product and the definition of the conjugate transpose, $A^{†}$, which we learned about last time.

The general strategy is to start with the eigenvalue equation, $Ax=\lambda x$, and show that the only way for the logic to hold for a symmetric/Hermitian matrix is if $\lambda =\bar{\lambda }$, which means $\lambda$ must be a real number.

For a detailed walkthrough of this elegant proof, let's turn back to Professor Gilbert Strang's lecture. He provides a clear, step-by-step demonstration.

Let's summarize the key steps of the proof you just saw:

1. Start with the eigenvalue equation: $Ax=\lambda x$. At this stage, we must assume $\lambda$ could be complex and $x$ could be a vector with complex entries.
2. Take the conjugate transpose of the whole equation: $(Ax{)}^{†}=(\lambda x{)}^{†}$. This gives $x^{†}{A}^{†}=\bar{\lambda }{x}^{†}$.
3. Use the matrix property:
   • For a real symmetric matrix $A$, we have $A^{†}=(\bar{A}{)}^T=A^T=A$.
   • For a Hermitian matrix $A$, we have $A^{†}=A$ by definition.
     In both cases, the equation becomes $x^{†}A=\bar{\lambda }{x}^{†}$.
4. Create two expressions for the same quantity:
   • Multiply the original equation, $Ax=\lambda x$, on the left by $x^{†}$:
     $x^{†}Ax=x^{†}(\lambda x)=\lambda (x^{†}x)$
   • Multiply the transposed equation, $x^{†}A=\bar{\lambda }{x}^{†}$, on the right by $x$:
     $x^{†}Ax=(\bar{\lambda }{x}^{†})x=\bar{\lambda }(x^{†}x)$
5. Equate and conclude: We now have two expressions for $x^{†}Ax$. Setting them equal gives:
   $\lambda (x^{†}x)=\bar{\lambda }(x^{†}x)$
   As Prof. Strang explained, $x^{†}x=\Vert x{\Vert }^2$ is the squared length of the vector. Since an eigenvector $x$ cannot be the zero vector, $x^{†}x$ is a positive real number. We can safely divide both sides by it.
   $\lambda =\bar{\lambda }$
   A number that is equal to its own complex conjugate must be real. The proof is complete!

The following resource provides a concise, written version of the proof for Hermitian matrices. Notice that the logic is identical.

2. Physical Motivation: Observables in Quantum Mechanics

So, why is this property so critically important for your goal of learning quantum mechanics?

In quantum theory, any quantity that you can physically measure—like energy, position, or spin—is called an observable. Each observable is represented by a specific Hermitian operator (which can be thought of as a Hermitian matrix). The possible results of a measurement are the eigenvalues of that operator.

Since the outcome of any real-world measurement is a real number (e.g., you measure an energy of 2.5 electron-volts, not $2.5+1.4i$ eV), it is a physical requirement that the eigenvalues of these operators must be real. Our proof shows that Hermitian matrices are precisely the right tool for the job.

The following video and text explain this connection from a physics perspective.

The argument in the video is essentially the reverse of our proof: it starts with the physical requirement of real eigenvalues and shows this implies the operator must be Hermitian. This reinforces the deep link between the mathematical property and the physical reality it describes.

This next article reinforces this same key idea.

Conclusion

In this lesson, we moved from stating a property to proving it and understanding its profound implications. This is a significant step in your journey towards understanding the mathematical foundations of both deep learning and quantum mechanics.

Key Takeaways:

• The eigenvalues of both real symmetric matrices and complex Hermitian matrices are always real numbers.
• The proof relies on starting with $Ax=\lambda x$, using the property $A=A^{†}$, and showing that this forces $\lambda =\bar{\lambda }$.
• A crucial step in the proof is that for any non-zero complex vector $x$, the inner product $x^{†}x$ is a positive real number, representing the vector's squared length.
• This mathematical property is a physical necessity in quantum mechanics, where measurable quantities (observables) are represented by Hermitian operators and measurement outcomes (eigenvalues) must be real.

Preview of the Next Lesson:

We've now established a crucial fact about the eigenvalues of symmetric/Hermitian matrices. But what about their eigenvectors? In the next lesson, we will prove the second major property of these matrices: their eigenvectors corresponding to distinct eigenvalues are orthogonal. This will lead us directly to one of the most important theorems in all of linear algebra: The Spectral Theorem, which states that every symmetric/Hermitian matrix can be diagonalized by a basis of orthonormal eigenvectors.