5. If N is an elementary n x n nilpotent matrix, i.e., N^k =...

5. If N is an elementary n x n nilpotent matrix, i.e., $N^k = 0$, $N^{k-1} \ge 0$. Prove that N do not have a squared root, i.e., there does not exist a matrix A such that $A^2 = N$. 6. Describe all possible Jordan forms of a nilpotent endomorphism over a vector space of dim V ? 6.

Transcript

00:01 A nilpotent matrix, a square matrix with the power that is the zero matrix.

00:07 That is, there is some natural number m, which a raised to the m power that is a in a times itself, m times equal zero matrix.

00:19 In part a, we are going to check that the given matrices are nil potent.

00:25 In part b, we show that a matrix similar to a nil potent matrix is also nil potent.

00:29 In per c we show that a nilpotent matrix is not a revertible.

00:35 And finally in part d we show that the only eigenvalue of a nilpotent matrix is 0.

00:43 So let's start by solving par a, that is, we are going to prove that the matrix a and matrix b are nilpotent.

00:57 So let's put a again here.

00:59 A is 0, 2, negative 3, 0, 0, 4, 0 ,000.

01:15 What we got to do is to find successive powers of a up to the time we get zero matrix.

01:24 So we get a square, a square is a times a, that is 0 ,2, negative 3, 00, 4, 000 times itself, 0 to negative 3, 0, 0 4, and 0 ,000.

01:43 And we get as result 0 0 0 times 2 is 0 to 0 and negative 3 then 0 0 0 times negative 3 is 8 and negative 3 times 0 is 0.

02:04 Then second row will be 0 0 0 0 0 0.

02:11 0 0 0 0 0 times 93 0 0 times 4 times 0 0 so we get 0 again then 0 again and 0 again so we don't get at the second power the zero matrix we are very close to so a cube is now a square times a that is a square found here here, 08, 0000, 0 ,000, 0 ,000 times a, 02, negative 3, 0, 04, and 000.

03:05 And we get 0, 0, and then 0.

03:13 And because the next two rows are 0, get 0.

03:18 The next two rows, so it's 0 and 0 matrix.

03:21 So a cube is equal to 0 and then a is nilpotent for m equal 3.

03:48 Now let's see b which is the matrix 3, net to 3 to 1 in each row and calculate b square.

04:12 So b square is net 2 to 1, 93 to 1, 903 to 1, and 93 to 1.

04:19 1 times itself and that is negative 3 times negative 3 is 9 2 times negative 6 and negative 3 then we have negative 2 is negative 6 4 2 then negative 3 plus 2 okay the second row will be you can see that what we did in the first row that is the inner product of negative 3 to 1 with negative 3, negative 3, negative 3, net 2, 9, 3, and with 2, 2, 2, and with 1, 1, 1.

05:13 It's going to be repeated in the next 2 rows of the product matrix.

05:20 That is because we have the same vector now here in the second row of b.

05:26 So we know the expressions are going to be the same ones we got in the first row.

05:34 So we get this and we get this.

05:43 Negative 6 plus 4 plus 2 and negative 3 plus 2 plus 1.

05:49 This is the product matrix.

05:51 And now we see that in the first row, we do the calculations and that would be the same calculations in the next two rows.

05:59 So here this value is 0...

Question

5. If N is an elementary n x n nilpotent matrix, i.e., $N^k = 0$, $N^{k-1} \ge 0$. Prove that N do not have a squared root, i.e., there does not exist a matrix A such that $A^2 = N$. 6. Describe all possible Jordan forms of a nilpotent endomorphism over a vector space of dim V ? 6.

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