Introduction

The study of linear algebra is fundamental in both theoretical and applied mathematics. One of the key properties of a real symmetric matrix is that it possesses real eigenvalues. This property is not trivial and requires a careful proof. In this article, we will explore how to prove the existence of eigenvalues for a real symmetric matrix using only basic algebraic knowledge and the Fundamental Theorem of Algebra.

Understanding Real Symmetric Matrices

A real symmetric matrix A is a square matrix that satisfies the condition A AT. This means that A is equal to its transpose. Such matrices have the remarkable property that they are orthogonally diagonalizable, meaning there exists an orthogonal matrix Q such that QTAQ D, where D is a diagonal matrix containing the eigenvalues of A.

Proof Using the Characteristic Polynomial

The characteristic polynomial of a matrix A is defined as P(λ) det(A - λI), where I is the identity matrix. The roots of this polynomial are the eigenvalues of A. For a n x n matrix, the characteristic polynomial is a polynomial of degree n in λ. The Fundamental Theorem of Algebra guarantees that this polynomial has exactly n roots in the complex plane, counting multiplicity.

Step-by-Step Proof

Consider a real symmetric matrix A. Its characteristic polynomial is given by:

P(λ) det(A - λI)

By the Fundamental Theorem of Algebra, the polynomial P(λ) has at least one complex root. Since P(λ) is a polynomial with real coefficients, the complex roots must occur in conjugate pairs.

Now, consider the specific case of a real symmetric matrix A. We can use the fact that the matrix is symmetric to show that all eigenvalues are real.

Let λ be an eigenvalue of A with corresponding eigenvector x. Then, by definition:

Ax λx

Multiplying both sides by xT, we get:

xTAx λxTx

Since A is symmetric, A AT, so:

xTAx xTAx λxTx

Consider the inner product xTAx, which is a real number because the matrix A is real and symmetric. Therefore, xTx is also a real number.

We can now write:

λ (xTAx)/(xTx)

Since both the numerator and the denominator are real, λ must be real.

Using the Rayleigh Quotient

Another approach to proving the existence of eigenvalues for real symmetric matrices is through the Rayleigh quotient. The Rayleigh quotient for a matrix A and a vector x is defined as:

r (xTAx)/(xTx)

The largest eigenvalue is the maximum of r over all non-zero vectors x, and the smallest eigenvalue is the minimum of r over all non-zero vectors x. The other eigenvalues lie in between these two values.

Conclusion

In conclusion, proving that a real symmetric matrix possesses eigenvalues using basic algebraic knowledge and the Fundamental Theorem of Algebra is a straightforward process. The Fundamental Theorem of Algebra guarantees the existence of eigenvalues, while the properties of real symmetric matrices ensure that these eigenvalues are real. By using methods like the characteristic polynomial and the Rayleigh quotient, we can further demonstrate and understand the properties of these matrices.