Chapter 7.6, Problem 1TFR

True-False Review

For Questions (a)-(l), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem in fact from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.

(a) Every eigenvector is a generalized eigenvector.

(b) The number of Jordan blocks in the Jordan canonical form of a matrix $A$ is the number of linearly independent eigenvectors of $A$.

(c) For every square matrix $A$, there is a unique invertible matrix $S$ such that $S^{-1}AS$ is in canonical form.

(d) If $J_1$ and $J_2$ are $n\times n$ matrices in Jordan canonical form, then the matrix $J_1+J_2$ is in Jordan canonical form.

(e) A generalized eigenvector of $A$ corresponding to an eigenvalue $\lambda$ is a member of the null space ${\left(A-\lambda I\right)}^p$ for some positive integer $p$.

(f) The dimension of $K_{\lambda }$, the vector space of generalized eigenvectors corresponding to the number of Jordan blocks corresponding to $\lambda$ in the Jordan canonical form of $A$.

(g) Every square matrix $A$ is similar to a matrix $J$ in Jordan canonical form.

(h) If $J_1$ and $J_2$ are $n\times n$ matrices in Jordan canonical form, then the matrix $J_1{J}_2$ is in Jordan canonical form.

(i) The size of a Jordan block is equal to the number of vectors in the corresponding cycle of generalized eigenvectors of $A$.

(j) If $A$ is an $n\times n$ matrix with no cycles of generalized eigenvectors of length $p\ge 2$, then $A$ is diagonalizable.

(k) Similar matrices must have the same Jordan canonical form, up to rearrangement of the Jordan blocks.

(l) If $J$ is in Jordan canonical form and $r$ is a scalar, then the matrix $rJ$ is in Jordan canonical form.