Eigenvectors
$Ax$ parallel to $x$
What does it mean to be parallel? ⇒ $Ax=\lambda x$
Example 1.1 $\lambda=0$
$\lambda=0$ 이면, Eigen vector들은 $A$의 Null Space가 된다. 즉 $A$가 non singular면 $\lambda=0$인 Eigen Value를 가진다.
Example 1.2 Projection
- For any $x$ in plane , $Px=x$.
- $\lambda =1$ and the whole plane becomes eigen vector
- For any $x\perp plane$ , $Px=0$
- eigen value becomes 0
⇒ Therefore, eigen value of projection matrix becomes 0 and 1.
Example 1.3 Permutation
$A=\begin{bmatrix}0&1\\1&0\end{bmatrix}$
For $x=\begin{bmatrix}1\\1\end{bmatrix}, Ax=x$ , $\lambda=1$
For $x=\begin{bmatrix}-1\\1\end{bmatrix}, Ax=-x,\lambda=-1$
Remark 1.4 (Sum of eigenvalues)
sum of diagonals of matrix = sum of eigenvalues
Remark 1.5 (Solving eigenvalues)
$Ax=\lambda x$
$(A-\lambda I)x=0$
For $x$ to exist, $A-I\lambda$ should be singular matrix.
Which means, $det(A-\lambda I)=0$
⇒ Find $\lambda$
Example 1.6 (Symmetric Matrix)
$A=\begin{bmatrix}3&1\\1&3\end{bmatrix}$
$det(A-\lambda I)=(3-\lambda)^2-1=\lambda^2-6\lambda+8=0$
- Observe that 6 is the trace, 8 is the determinant of A
$\lambda = 2,4$
Example 1.7 ( Adding $I$)
Example 1.6 의 symmetric matrix는 Example 1.3의 matrix에 $3I$를 더한 것인데, Eigen Value도 3씩 증가한 것을 확인할 수 있다.
Then is it true that eigen values of $A$ and $B$, then $A+B$ has eigenvalues $\alpha+\beta$?
This is false because the corresponding eigen vecotrs may be different.
$Ax=\alpha x$, $By=\beta y$ but it still holds for $I$
Example 1.8 ( Rotation Matrix )
$Q=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$
Trace = sum of eigenvalues
det = multiple of eigenvalues
This Matrix rotates vector 90 degrees. What vector stays still after 90 degrees rotation?
$det(Q-\lambda I)=\lambda^2+1=0$
$\lambda=i,=i$
Example 1.9 (Transpose)
Transpose Matrix has same eigen values.
$(A-\lambda I)^T=A^T-\lambda I$
Therefore,
$det(A-\lambda I)=det((A-\lambda I)^T)=det(A^T-\lambda I)$
Therefore, they have the same eigen values.
'Math > Linear Algebra' 카테고리의 다른 글
Diagonlizable Matrix (0) | 2021.04.25 |
---|---|
Cramer's Rule, Inverse Matrix, Volume (0) | 2021.02.28 |
Cofactors and Determinants (0) | 2021.02.19 |
Determinant (0) | 2021.02.12 |
Orthonormal basis (0) | 2021.02.10 |