Science:Math Exam Resources/Courses/MATH307/April 2010/Question 02 (c)
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Question 02 (c)
A real matrix is brought into its Jordan canonical form
by a matrix , i.e. . Answer the questions below, substantiate all your statements.
(c) What is the dimension of the kernel of (I being the 5x5 identity matrix)?
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Science:Math Exam Resources/Courses/MATH307/April 2010/Question 02 (c)/Hint 1
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The kernel of where
Since , then .
This means to find , 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. for the eigenvalue -1 is equal to 2.