The Spectral Theorem for Real Symmetric Matrices

Matrices play a fundamental role in the field of linear algebra. Among them, the class of real symmetric matrices holds a unique status due to their properties and the uncanny simplicity in their eigenvalues and eigenvectors. In this article, we delve into the reasons why all eigenvalues of a real symmetric matrix are real and all eigenvectors are perpendicular to each other. We will explore the consequences of these unique properties and provide a detailed proof of the spectral theorem for real symmetric matrices.

Properties of Real Symmetric Matrices

A matrix A is considered real symmetric if it satisfies the condition A AT, where AT denotes the transpose of A. This implies that all elements Aij are real, ensuring the matrix is entirely composed of real numbers.

Real Eigenvalues of Real Symmetric Matrices

One of the most important properties of real symmetric matrices is that all their eigenvalues are real. This is a direct consequence of the real nature of the matrix and its symmetric property. To prove this, we start with an eigenvector v corresponding to an eigenvalue λ of a real symmetric matrix A. We have the equation:

Av λv

By taking the complex conjugate of both sides, we obtain:

A* v λ*v

Since A is real, A* AT A. Hence, we have:

Av λ*v

Multiplying both sides by vT and using the inner product, we get:

vTAv λ*vTv

Since A AT, we replace Av with vTAv:

vTAv vTAv

Combining this with the previous equation, we obtain:

λvTv λ*vTv

Rearranging, we get:

λ λ*

Since vTv ≠ 0 for any non-zero eigenvector v, this implies that λ is a real number.

Orthogonal Eigenvectors

Another intriguing property of real symmetric matrices is that eigenvectors corresponding to distinct eigenvalues are orthogonal. This can be shown using the fact that if u and v are eigenvectors corresponding to distinct eigenvalues λ and μ, then:

Au λu and Av μv

By taking the inner product of u with both sides of the first equation, we get:

uTAu λuTu

Using the symmetry of A, we have:

uTAu ATuTu AuTu

Therefore, λuTu AvTu.

Similarly, for the second equation:

vTAv μvTv

Using the symmetry of A, we get:

vTAv ATvTv AvTv

Therefore, μvTv AuTv.

Combining these, we obtain:

λuTv μuTv

Since λ ≠ μ, it follows that:

uTv 0

Thus, eigenvectors corresponding to distinct eigenvalues are orthogonal.

Orthogonal Spectral Decomposition

From the above properties, we can deduce that a real symmetric matrix A can be orthogonally decomposed into:

A UΛUT

where U is an orthogonal matrix with columns being the orthonormal eigenvectors of A, and Λ is a diagonal matrix with the corresponding eigenvalues on the diagonal. This orthogonal spectral decomposition is a powerful tool in linear algebra and is the basis for many numerical algorithms used in scientific computations.

Conclusion

The spectral theorem for real symmetric matrices is a cornerstone of linear algebra, providing us with the assurance that the eigenvalues are real and the eigenvectors are orthogonal. This theorem not only simplifies many problems in linear algebra but also extends to various fields such as physics, engineering, and data science. Understanding these properties is crucial for anyone working with matrices and linear transformations.

Keywords: real symmetric matrix, eigenvalues, eigenvectors, spectral theorem