Problem 11 - Differential Equations and Linear Algebra

Question

The linear transformation \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) with matrix \(A=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{array}\right]\) takes vectors \((x, y, z)\) in \(\mathbb{R}^{3}\) and moves them to the corresponding point \((0, y, 0)\) on the \(y\) -axis. By arguing geometrically, determine all eigenvalues and eigenvectors of \(A\).

Step-by-Step Solution

Answer

The eigenvalues and eigenvectors of the matrix A are: Eigenvalue 1: Eigenvectors of the form \((0, y, 0)\) with \(y \neq 0\). Eigenvalue 0: Eigenvectors of the form \((x, 0, z)\) with either \(x \neq 0 \) or \( z\neq 0 \).

Step 1: Eigenvectors on the y-axis

Every point on the y-axis, besides the origin, is an eigenvector corresponding to eigenvalue 1. This is because the transformation moves them to the same point, thus scaling by a factor of 1. These eigenvectors are of the form \((0, y, 0)\), where \( y\neq 0 \).

Step 2: Eigenvectors orthogonal to the y-axis

Points \( (x, 0, z) \), where \(x \neq 0\) or \(z \neq 0\), will be moved to the origin by the transformation. Hence, they will be scaled by a factor of 0. Thus, eigenvectors of the form \( (x, 0, z) \) will have an eigenvalue of 0.

Step 3: Summary of eigenvalues and eigenvectors

The eigenvalues and eigenvectors of the matrix A are as follows: Eigenvalue 1: Eigenvectors of the form \((0, y, 0)\) with \(y \neq 0\). Eigenvalue 0: Eigenvectors of the form \((x, 0, z)\) with either \(x \neq 0 \) or \( z\neq 0 \).

Key Concepts

Eigenvalues

When discussing linear transformations and matrices, eigenvalues are special numbers that indicate how a transformation scales its eigenvectors. For the given matrix\[A=\left[\begin{array}{ccc}0 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0\end{array}\right]\]eigenvalues can be found by solving the characteristic equation \(\det(A - \lambda I) = 0\). In simpler terms, eigenvalues represent the amount by which eigenvectors are stretched or compressed.

For this matrix, there are two important eigenvalues:

• Eigenvalue 1: This indicates vectors remain unchanged in magnitude along the y-axis, as the transformation maps them directly onto themselves.
• Eigenvalue 0: This shows vectors are compressed (or moved) to the origin, meaning they disappear in magnitude compared to their original state.

Understanding eigenvalues aids in grasping how transformations affect vectors in terms of stretching or collapsing them, which is essential for interpreting the geometric effects in vector spaces.

Eigenvectors

Eigenvectors inform us about directions in a vector space that remain unchanged by a transformation, except for scalar multiplication. In our context, each eigenvalue corresponds to a set of eigenvectors that reveals its unique direction. Let's consider the matrix given:\[A=\left[\begin{array}{ccc}0 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0\end{array}\right]\]

For eigenvalue 1, all vectors along the y-axis (except the origin) are eigenvectors, as they map onto themselves scaled by 1 during transformation. These vectors can be represented as:

• Eigenvectors of the form \((0, y, 0)\) with \(y eq 0\).

On the opposite side, eigenvalue 0 relates to vectors orthogonal to the y-axis.

• These eigenvectors have the form \((x, 0, z)\) where either \(xeq 0\) or \(z eq 0\), indicating they are squished down to the origin.

Understanding eigenvectors provides insight into the directions along which the transformation behaves scalarly, keeping those directions intact.

Geometric Interpretation

The geometric interpretation of eigenvalues and eigenvectors gives us a visual understanding of transformations. For matrix \(A = \left[\begin{array}{ccc}0 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0\end{array}\right]\), observing its effects geometrically can be fascinating.

Picture a 3D space where each direction (axis) corresponds to a dimension. The matrix applies the following transformation:

• Vectors along the y-axis remain fixed in direction, reflecting eigenvalue 1.
• Vectors along the x-z plane collapse to a flat point at the origin, illustrating eigenvalue 0.

This interplay depicts how the linear transformation compresses and retains certain parts of its space.

The y-axis represents a line of stability and constancy, while any vector off this line is essentially removed. This intuitive spatial thought aids deeper comprehension in how linear transformations affect overall structure within their vector spaces.