Question

Suppose the matrix $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ has a repeated eigenvalue \(\lambda_{1}\) and the associated eigenspace is one- dimensional. Let \(\mathbf{x}\) be a \(\lambda_{1}-\) eigenvector of \(A .\) Show that if \(\left(A-\lambda_{1} I\right) \mathbf{u}_{1}=\mathbf{x}\) and \(\left(A-\lambda_{1} I\right) \mathbf{u}_{2}=\mathbf{x},\) then \(\mathbf{u}_{2}-\mathbf{u}_{1}\) is parallel to \(\mathbf{x}\). Conclude from this that all vectors \(\mathbf{u}\) such that \(\left(A-\lambda_{1} I\right) \mathbf{u}=\mathbf{x}\) define the same positive and negative half-planes with respect to the line \(L\) through the origin parallel to \(\mathbf{X}\).

Short answer

Question: Show that if (A - λ₁I)u₁ = x and (A - λ₁I)u₂ = x, then the vector (u₂ - u₁) is parallel to x. Furthermore, conclude that any vector u satisfying (A - λ₁I)u = x defines a positive or negative half-plane with respect to line L, which is parallel to x and passes through the origin. Answer: To show that (u₂ - u₁) is parallel to x, we subtract the given equations to get (A - λ₁I)(u₂ - u₁) = 0. This implies that (u₂ - u₁) is in the null space of (A - λ₁I) and is parallel to x. Consequently, any vector u satisfying (A - λ₁I)u = x will define the same positive or negative half-plane with respect to the line L through the origin, which is parallel to x, as all such vectors lie in the same half-plane defined by line L, either in the positive or the negative direction of x.

Step-by-step solution

Apply the matrix to the difference of the vectors u₂ and u₁

We are given that (A - λ₁I)u₁ = x and (A - λ₁I)u₂ = x. Let's subtract these two equations to get an expression for the matrix applied to u₂ - u₁. $$ (A - \lambda_1 I)u_2 - (A - \lambda_1 I)u_1 = x - x $$

Simplify the equation

Now, let's simplify the above equation using the associativity and distributive properties of matrices: $$ (A - \lambda_1 I)(u_2 - u_1) = 0 $$

Interpret the result

The equation (A - λ₁I)(u₂ - u₁) = 0 tells us that the vector u₂ - u₁ is in the null space of (A - λ₁I). Since x is an eigenvector of A corresponding to eigenvalue λ₁, (u₂ - u₁) is parallel to x.

Conclude the properties of the half-planes

Since u₂ - u₁ is parallel to x, any vector u satisfying (A - λ₁I)u = x will define the same positive or negative half-plane with respect to the line L through the origin, which is parallel to x. This is because the vector u₂ - u₁ is a scalar multiple of x, meaning all such vectors u lie in the same half-plane defined by line L, either in the positive or the negative direction of x.

Key concepts

Eigenvalues

Understanding eigenvalues is fundamental in linear algebra. An eigenvalue, denoted normally as \( \lambda \), is a scalar that indicates how much a corresponding eigenvector is stretched during a linear transformation represented by a matrix.

• When you have a matrix \( A \), and a non-zero vector \( \mathbf{v} \), if \( A \mathbf{v} = \lambda \mathbf{v} \), then \( \lambda \) is the eigenvalue associated with \( \mathbf{v} \).
• In some cases, like the one in the original exercise, eigenvalues can be repeated. This is known as having a multiplicity greater than one.
• An important insight from repeated eigenvalues is that the transformation effect associated with that eigenvalue is identical across multiple dimensions or directions.

Eigenvalues play a major role in matrix theory and applications such as stability analysis, vibrations, and many other fields.

Eigenvectors

Eigenvectors are intimately connected with eigenvalues, representing the direction of the transformation. For a matrix \( A \), if \( \mathbf{v} \) is a non-zero vector such that \( A \mathbf{v} = \lambda \mathbf{v} \), \( \mathbf{v} \) is the eigenvector associated with eigenvalue \( \lambda \).

• They provide critical insights, showing directions in which transformations do not occur.
• The lines formed by eigenvectors through the origin are significant for visualizing transformations.
• In the concept addressed in the exercise, if two vectors \( \mathbf{u}_2 \) and \( \mathbf{u}_1 \) are such that their difference is an eigenvector \( \mathbf{x} \), it highlights a parallel relationship indicating consistent directional influence of the matrix transformation.

Eigenvectors remain stable under the linear transformation associated with their eigenvalue but may be stretched or compressed, which is invaluable for various applications, including computer graphics and quantum mechanics.

Matrix Theory

Matrix theory forms the backbone of understanding complex linear transformations. It involves the systematic study of arithmetic and structural properties of matrices.

• Matrices like \( A \) in the exercise represent linear transformations that apply operations to vector spaces.
• The matrix \( (A - \lambda I) \), which involves subtracting an eigenvalue \( \lambda \) times the identity matrix, is critical in eigenvalue problems. It helps isolate or neutralize transformations to understand behavior under specific conditions.
• Matrices help in determining spaces such as null spaces and eigenspaces, significant in solving linear equations, performing dimensional analysis, and finding vector space orientations.

By studying matrix theory, one can learn to manipulate and predict outcomes of linear transformations, making it integral in areas like machine learning, systems theory, and physics.

Null Space

The null space, or kernel, of a matrix is an important concept, representing all the vectors that, when transformed by a matrix, result in the zero vector. For a matrix \( A \), the null space consists of all solutions \( \mathbf{v} \) such that \( A \mathbf{v} = 0 \).

• In the context of the exercise, the null space was found from the equation \( (A - \lambda_1 I)(u_2 - u_1) = 0 \).
• This equation suggests that \( \mathbf{u}_2 - \mathbf{u}_1 \) lies in the null space of \( (A - \lambda_1 I) \).
• Understanding the null space helps determine the linear dependencies and freedom within a transformation.

The concept is vital in solving systems of equations, understanding vector space dimensions, and applications like signal processing, where transformations need to nullify specific components.