Eigenvectors and eigenvalues

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Prerequisites:

• Linear transformations and matrices
• The determinant
• Change of basis vector

Eigen Vectors:

When a space is transformed, using a linear Transformation, some vectors retain their original direction, they just get stretched or squished by some factor. These vectors are called Eigen Vectors and the values by which they are stretched or squished is called Eigen Value.

If the direction of a vector is flipped then the eigen value of that vector for that transformation will be negative.

Identity matrix:

The matrix with 1 along the diagonal and 0 everywhere else. Denoted by I

Why might this be useful?

The eigen vector can be thought of as the axis around which a space is transformed.

Formula for Eigen Vector:

A$\vec{v}$ = $\lambda \vec{v}$ where, A is the Transformation $\vec{v}$ is the eigen vector $\lambda$ is the eigen value

Interpretation:

Applying a transformation over the eigen vector only scales that vector by the eigen value. (This is just the exact definition of Eigen vectors).

since $\lambda$ is a scalar, it can be converted to matrix multiplication, think of it as a transformation where the basis vectors are scaled by $\lambda$, This can be considered as $\lambda$ multiplied to the Identity matrix.

so can be written as A$\vec{v}=(\lambda I)\vec{v}$ $A\vec{v}-(\lambda I)\vec{v}=0$ $(A-\lambda I)\vec{v}$ =0

This means the Transformation is squishing the vector in a lower dimension.

Info

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so we strive to find the value of $\lambda$ which will get the determinant of ($(A-\lambda I)\vec{v}$) to 0.

To find the eigen vectors, compute (A-$\lambda I$) and for which vector v, when the transformation is applied the result is 0.

A 2D transformation may not always have an eigen vector or eigen value, like a rotation of $90^o$ to the left, all the vectors have changed their direction. if we compute the eigen value it will be $-i$ and $i$.

There can be many eigen vector and many eigen values for a transformation

Consider the trasnformation that scales everything by a scalar, without changing the direction.