Why Are Eigenvectors Considered the "Direction" of Linear Transformations?

Eigenvectors are really interesting because they show us the true "direction" of transformations! 🌟 Let’s break it down:

What are Eigenvectors? An eigenvector, which we can call $\mathbf{v}$ , meets the equation $A\mathbf{v} = \lambda\mathbf{v}$ . In this, $A$ is like a special matrix that describes how we change something, and $\lambda$ is a number called the eigenvalue.
Scaling: This equation means that when we transform an eigenvector, it becomes a bigger or smaller version of itself. But the direction doesn't change!
Why They Matter: Eigenvectors show us the stable directions in a transformation. This helps us understand and analyze different systems better! 🎉

Eigenvectors are really interesting because they show us the true "direction" of transformations! 🌟 Let’s break it down:

What are Eigenvectors? An eigenvector, which we can call $\mathbf{v}$ , meets the equation $A\mathbf{v} = \lambda\mathbf{v}$ . In this, $A$ is like a special matrix that describes how we change something, and $\lambda$ is a number called the eigenvalue.
Scaling: This equation means that when we transform an eigenvector, it becomes a bigger or smaller version of itself. But the direction doesn't change!
Why They Matter: Eigenvectors show us the stable directions in a transformation. This helps us understand and analyze different systems better! 🎉

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