Science:Math Exam Resources/Courses/MATH307/April 2010/Question 02 (c)

MATH307 April 2010

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Question 02 (c)
A real matrix $A_{5\times 5}$ is brought into its Jordan canonical form $B={\begin{pmatrix}1&1&0&0&0\\0&1&0&0&0\\0&0&0&0&0\\0&0&0&-1&0\\0&0&0&0&-1\end{pmatrix}}$ by a matrix $S_{5\times 5}$, i.e. $\displaystyle A=SBS^{-1}$. Answer the questions below, substantiate all your statements. (c) What is the dimension of the kernel of $\displaystyle A+I$ (I being the 5x5 identity matrix)?

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Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The kernel of $A+I=N(A+I)$ where $N(A+I)=\left\{x|(A+I)x=0\right\}$ Since $(A+I)x=0$, then $(A-(-1)I)x=0$. This means to find $\dim(N(A+I))$, we just need to find the dimension of the nullspace of A with the eigenvalue -1. From matrix B, we can see that the eigenvalue -1 appears twice, and that they are not from the same Jordan block. If they were from the same Jordan block, then there would be a 1 shown at entry (4,5). Since it appears twice, we know that the dimension of the eigenspace (i.e. $N(A-(-1)I)$ for the eigenvalue -1 is equal to 2. Therefore, $\dim(N(A+I))=2$