calculating eigenspaces

Solution 1

You can find the Eigenspace (the space generated by the eigenvector(s)) corresponding to each Eigenvalue by finding the kernel of the matrix $A-\lambda I$. This is equivalent to solving $(A-\lambda I)x=0$ for $x$.

In your case:

For $\lambda =1$ the eigenvectors are $(1,0,2)$ and $(0,1,-3)$ and the eigenspace is $gen\{(1,0,2);(0,1,-3)\}$ For $\lambda =2$ the eigenvector is $(0,-2,5)$ and the eigenspace is $gen\{(0,-2,5)\}$

Solution 2

Denote $A$ your matrix. To find the eigenspace of $\lambda$ solve for $X=(x,y,z)^T$ the equation $$AX=\lambda X$$

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Updated on August 01, 2022