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The word *eigen*, having a German origin, means *characteristics*. The eigenvalues and eigenvectors give the characteristic, but of what? Let’s understand it through a geometric example.

The linear transformations such as scaling, rotation, and shearing can be expressed using matrices. For example, by applying a vertical scaling of +2 to every vector of a square, will transform the square into a rectangle. In the same way, by applying a horizontal shear to the square, it becomes a parallelogram.

Note that during these transformations, some of the vectors remain on the same line (span) as they were earlier. As shown in the figure,

- The horizontal vector remains unchanged (same direction, same length).
- The vertical vector has same direction, but doubled in length.
- The diagonal vector has changed its angle (direction) as well as length.

Note that, in the above figure, after vertical scaling of +2, every vector’s (except horizontal and vertical ones) direction has changed. These two vectors are special and are the characteristic of this particular transform. Hence, these are called **eigenvectors**.

An eigenvector is a vector, which after applying the linear transformation, stays in the same span i.e. changes by only a scalar factor.

The **eigenvalue** is how much the eigenvectors are transformed (stretched or diminished).

- The horizontal vector’s length remains same, thus have an eigenvalue of +1.
- The vertical vectors’ length doubled, thus have an eigenvalue of +2.

In the same way, the horizontal shear transformation to square gives only one eigenvector (horizontal one) having an eigenvalue of +1.

In 180 degree rotation of square, all vectors are still laying on the same span, but their direction is reversed. Hence, all vectors are eigenvectors, having an eigenvalue of -1.

In case of 3d rotation transformation of cube, the eigenvector gives the axis of rotation.

Suppose we have a transformation matrix `A`

and we apply this transformation to vector `x`

. This will be equivalent to stretching (or diminshing) the vector `x`

by a scalar factor `λ`

.

where `I`

is the identity matrix.

The above euqation has a non-zero solution iff the determinant of the matrix `(A − λI)`

is zero i.e.

Evaluating this determinant gives the characteristic polynomial i.e.

\[\text{if } A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\] \[\begin{vmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} - \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} \end{vmatrix} = 0\] \[\lambda ^ 2 - (a+d)\lambda + ad - bc = 0\]The solution of this equation gives the eigvenvalues. Put these eigenvalues in the original expression to get their corresponding eigenvectors.

**References:**

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