Basic Method

When learning the eigenvalue and eigenvector, we are commonly given an equation at the beginning $$ AX = \lambda X \tag{1} $$ Where

  • A is the matrix
  • X is the eigenvector
  • $\lambda$ is the eigenvalue

Take an example

Q: Find the eigenvalues and eigenvectors of the matrix $$ A = \begin{bmatrix} 1&1&-2\ -1&2&1\ 0&1&-1 \end{bmatrix} $$ A: To get the answer, firstly apply the equation (1) : $$ AX = \lambda X \tag{1} $$

Before solving the equation, there might be something awkward, since on the right-hand side of the equation, it’s matrix-vector multiplication; while on the left-hand side, it’s the scalar-vector multiplication.

We can rewrite the right-hand side as some kind of matrix-scalar multiplication, namely, using the identity matrix which has the effect of scaling any vector by $\lambda$

I is the identity matrix here, form like:

Hence: $$ (\lambda I-A)X=0 \tag{2} $$ In order to get a non-trivial solution, we get : $$ \left| \lambda I-A\right| = 0 \tag{3} $$ where equation (3) is also called characteristic function. Expanding equation (3): $$ \begin{bmatrix}\lambda -1&-1&2\1&\lambda -2&-1\0&-1&\lambda +1\end{bmatrix} = det(\lambda I -A)=\lambda^3 -2\lambda^2 -\lambda +2=0 \tag{4} $$ Solving equation(4) we got eigenvalues as: $$ \begin{cases}\lambda_1 = 2\ \lambda_2 = 1\ \lambda_3 = -1\end{cases} $$ Now we can find the related eigenvector respectively by taking eigenvalues back to equation (2): $$ \begin{bmatrix}1-2&1&-2\-1&2-2&1\0&1&-1-2\end{bmatrix}\begin{bmatrix}e_1\e_2\e_3\end{bmatrix}=0\ (when\ \lambda=2) $$ hence: $$ \frac{e_1}{-1}=\frac{-e_2}{3}=\frac{e_3}{-1} $$ Unless otherwise stated, the eigenvectors will always be presented in their ‘simplest’ form, so the matrix of the eigenvector can be written as: $$ \begin{bmatrix}1\3\1\end{bmatrix}\ (when\ \lambda =2) $$ The other two eigenvector can be calculated similarly.

What does it mean?

Start from definition

looking back to the equation (1) again: $$ AX=\lambda X \tag{1} $$

  • On the right hand side, A is a matrix, stands for a kind of linear transformation (rotation, stretching).
  • On the left hand side, $\lambda$ is a scalar
  • Matrix A multiply by x means applying a transformation on the vector X (rotation or stretching), and the effect of this kind of transformation is equal to the scalar $\lambda$ multiply by the vector X. (Only stretched)
  • By finding the eigenvalue and eigenvector, we can find out which vectors (eigenvectors) can be stretched only by the matrix, and what’s the extent to which it’s stretched (eigenvalue.)