Geometric Interpretation

While the equation $Av = \lambda v$ provides a precise algebraic definition, the true intuition behind eigenvalues and eigenvectors comes from geometry. Thinking of a matrix as a function that transforms space helps us see what these special vectors and scalars really represent.

Imagine you have a grid of points in a 2D plane. When you apply a matrix transformation, this grid can be stretched, compressed, rotated, or sheared. Most vectors on this grid will be knocked off their original direction. For example, a vector pointing northeast might end up pointing east after the transformation.

However, eigenvectors are the exceptions. They are the vectors that lie on lines that pass through the origin and do not change their direction during the transformation. The matrix only scales them, making them longer or shorter.

The Axes of Transformation

Think of eigenvectors as the primary axes of a transformation. They define the directions where the transformation's behavior is simplest: pure stretching or compressing.

Let's consider a transformation matrix $A$:

$$A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$$

This matrix scales anything along the x-axis by a factor of 2 and anything along the y-axis by a factor of 3.

• Vector on the x-axis: Take a vector $v_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$. Applying the transformation gives:

  $$Av_1 = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$$

  The resulting vector is $2 \times v_1$. It's still on the x-axis, just twice as long. Here, $v_1$ is an eigenvector, and its corresponding eigenvalue is $\lambda_1 = 2$.

• Vector on the y-axis: Now take a vector $v_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Applying the transformation gives:

  $$Av_2 = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 3 \end{pmatrix}$$

  The resulting vector is $3 \times v_2$. It's still on the y-axis but is now three times as long. So, $v_2$ is another eigenvector, with an eigenvalue of $\lambda_2 = 3$.

• An arbitrary vector: What about a vector that is not on these axes, like $v_3 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$?

  $$Av_3 = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$$

  The original vector $v_3$ pointed along the line $y=x$. The new vector $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ points in a completely different direction. Therefore, $v_3$ is not an eigenvector of this transformation.

The following plot illustrates this. The solid lines are the original vectors, and the dashed lines are the transformed vectors. Notice how the blue and green vectors stay on their original lines (they are eigenvectors), while the orange vector is rotated to a new direction.

The eigenvectors $v_1$ and $v_2$ are only scaled by the transformation. The non-eigenvector $v_3$ changes its direction.

The Meaning of the Eigenvalue (λ)

The eigenvalue $\lambda$ is the scaling factor. Its value tells us exactly how the eigenvector is stretched or shrunk.

• If $|\lambda| > 1$, the eigenvector is stretched.
• If $|\lambda| < 1$, the eigenvector is shrunk or compressed.
• If $\lambda = 1$, the eigenvector is unchanged by the transformation. It lies in a "line of stability."
• If $\lambda = 0$, the eigenvector is squashed to the origin. This means the transformation collapses the space along this eigenvector's dimension.
• If $\lambda$ is negative, the eigenvector is flipped to point in the opposite direction, in addition to being scaled by its magnitude. For instance, an eigenvalue of -2 means the vector points the other way and is twice as long.

What About Rotations?

Consider a matrix that rotates every vector by 90 degrees counter-clockwise:

$$R = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$

If you apply this transformation, every vector changes direction. A rotation matrix like this has no real eigenvectors because no vector (except the zero vector) keeps its direction. Its eigenvalues are complex numbers, which is a topic for more advanced studies. For now, it's enough to know that not all transformations have eigenvectors that can be drawn in standard 2D or 3D space.

In summary, the geometric view is a powerful way to understand this topic. Eigenvectors are the stable directions of a transformation, and eigenvalues quantify the amount of stretch or compression along those stable directions. This provides a fundamental description of what a matrix does to the space it acts upon.