Question

Let \(A\) be an \((n \times n)\) real symmetric matrix. Show that eigenvectors belonging to distinct eigenvalues are orthogonal. That is, if \(A \mathbf{x}_{1}=\lambda_{1} \mathbf{x}_{1}\) and \(A \mathbf{x}_{2}=\lambda_{2} \mathbf{x}_{2}\), where \(\lambda_{1} \neq \lambda_{2}\), then \(\mathbf{x}_{1}^{T} \mathbf{x}_{2}=0 .\) [Hint: Consider the matrix product \(\mathbf{x}_{1}^{T} A \mathbf{x}_{2}\), and use the symmetry of \(A\) to show that \(\left(\lambda_{1}-\lambda_{2}\right) \mathbf{x}_{1}^{T} \mathbf{x}_{2}=0\). You will also need to recall that if the matrix product of \(R\) and \(S\) is defined, then \((R S)^{T}=S^{T} R^{T}\).]

Short answer

In a real symmetric matrix, eigenvectors associated with distinct eigenvalues are orthogonal, which means that their dot product is equal to zero. This fact has been proven by manipulating the matrix product $𝑥_1^T A 𝑥_2$ using the properties of symmetric matrices, eigenvalues, and eigenvectors, and showing that $\langle \mathbf{x_1}, \mathbf{x_2} \rangle = 0$.

Step-by-step solution

Write the given information

We are given a real symmetric matrix \(A\) and eigenvectors \(\mathbf{x}_1\) and \(\mathbf{x}_2\) corresponding to distinct eigenvalues \(\lambda_1\) and \(\lambda_2\), respectively. This means that: $$ A\mathbf{x}_1 = \lambda_1\mathbf{x}_1 $$ and $$ A\mathbf{x}_2 = \lambda_2\mathbf{x}_2 $$

Consider the hint and compute the matrix product

As per the hint, we will compute the matrix product \(\mathbf{x}_1^T A \mathbf{x}_2\). This can be rewritten using the properties of eigenvectors as follows: $$ \mathbf{x}_1^T A \mathbf{x}_2 = (\lambda_1 \mathbf{x}_1)^T \mathbf{x}_2 $$ Now using the properties of matrix transpose, we get: $$ (\lambda_1 \mathbf{x}_1)^T \mathbf{x}_2 = \lambda_1 (\mathbf{x}_1^T \mathbf{x}_2) = \lambda_1 \langle \mathbf{x_1}, \mathbf{x_2} \rangle $$

Compute the product using the symmetry of \(A\)

Now we will compute the same matrix product \(\mathbf{x}_1^T A \mathbf{x}_2\) using the symmetry of \(A\): $$ \mathbf{x}_1^T A \mathbf{x}_2 = (\mathbf{x}_1^T A^T) \mathbf{x}_2 $$ Since \(A\) is symmetric, \(A^T = A\). Thus, this becomes: $$ \mathbf{x}_1^T A \mathbf{x}_2 = (\mathbf{x}_1^T A) \mathbf{x}_2 = (\mathbf{x}_1^T (\lambda_2 \mathbf{x}_2)) = \lambda_2 (\mathbf{x}_1^T \mathbf{x}_2) = \lambda_2 \langle \mathbf{x_1}, \mathbf{x_2} \rangle $$

Set the two expressions equal and solve for the dot product

Since both expressions are equal to \(\mathbf{x}_1^T A \mathbf{x}_2\), we can set them equal to each other: $$ \lambda_1 \langle \mathbf{x_1}, \mathbf{x_2} \rangle = \lambda_2 \langle \mathbf{x_1}, \mathbf{x_2} \rangle $$ Now, since \(\lambda_1 \neq \lambda_2\), we can factor out the dot product and solve for it: $$ (\lambda_1 - \lambda_2) \langle \mathbf{x_1}, \mathbf{x_2} \rangle = 0 $$ Since \(\lambda_1 \neq \lambda_2\), this implies that: $$ \langle \mathbf{x_1}, \mathbf{x_2} \rangle = 0 $$

Conclusion

We have shown that the dot product of eigenvectors corresponding to distinct eigenvalues is equal to zero. Therefore, eigenvectors associated with distinct eigenvalues are orthogonal.

Key concepts

Real Symmetric Matrix

A real symmetric matrix is a square matrix that is equal to its own transpose, meaning if you take the matrix and flip it over its diagonal, you end up with the same matrix. Mathematically, this property is denoted as:

• \[ A = A^T \]

Any real symmetric matrix has some fascinating properties that make it a central topic in linear algebra. For instance, its eigenvalues are always real. This is crucial because eigenvalues help us understand various properties of the matrix, such as stability in systems and important transformations.

Another property of real symmetric matrices is that there exists a set of orthogonal vectors (eigenvectors) that can diagonalize the matrix. This means we can transform the matrix into a diagonal form using these vectors, greatly simplifying many computations.
Understanding real symmetric matrices helps you grasp more complex linear algebra concepts. They frequently appear across scientific fields, from physics to computer science, because they naturally model so many real-world phenomena.

Orthogonality of Vectors

Orthogonality of vectors is a fundamental concept in linear algebra. Two vectors are said to be orthogonal if their dot product is zero:

• \( \mathbf{x}_1^T \mathbf{x}_2 = 0 \)

This means the vectors are perpendicular to each other in an n-dimensional space. Understanding orthogonality can greatly simplify problems in linear algebra, as orthogonal vectors are independent and often easier to work with.

In the context of real symmetric matrices, the orthogonality of eigenvectors corresponding to distinct eigenvalues is a key property. This allows us to break down complex matrix operations into simpler, independent operations.
Orthogonality is important not just theoretically, but also practically. In fields like machine learning and computer graphics, orthogonal vectors simplify algorithms and reduce computational complexity.

Linear Algebra Proofs

Linear algebra proofs are essential for validating and understanding the relationships and properties between matrices, vectors, and transformations. A proof demonstrates, using logical reasoning and mathematical concepts, that a statement or formula holds true.

A significant proof in linear algebra, related to real symmetric matrices, involves showing that eigenvectors associated with distinct eigenvalues are orthogonal. This proof benefits from leveraging the properties of matrix transposition and symmetry. By setting equations from these properties equal and solving for the dot product, one can clearly infer the orthogonality of the vectors.

• The hint given in the original problem, \( \mathbf{x}_1^T A \mathbf{x}_2 = (\lambda_1 - \lambda_2) \mathbf{x}_1^T \mathbf{x}_2 = 0 \), neatly guides you through this reasoning.

Proofs like these are pivotal as they reinforce theoretical understanding and provide clear reasoning that can be applied to many disciplines in science and engineering. They encourage students to form a structured thought process, building foundational skills that are necessary for tackling more complex mathematical challenges.