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Q16E

Expert-verifiedFound in: Page 323

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.**

**Rotation through an angle of** ${{\mathbf{180}}}^{{\mathbf{\xb0}}}$** in ${{\mathit{R}}}^{{\mathbf{2}}}$****.**

So, *A* is diagonalizable.

A matrix is known as diagonalizable if it can be written in the form of ${\mathit{A}}{\mathbf{=}}{\mathit{P}}{\mathit{D}}{{\mathit{P}}}^{\mathbf{-}\mathbf{1}}$.

Rotation through an angle of ${180}^{\xb0}$ will turn any vector *v* into *-v* , so role="math" localid="1659525298008" $A=-{l}_{2}$, and all vectors $v\in {R}^{2}$are eigenvectors of *A* with the corresponding eigenvalue being $\lambda =-1$. The very fact that $A=-{l}_{2}$ is already diagonal proves that it's diagonalizable.

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