Section VI.7

§ 7

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Problem VI.7.1

How many Jordan matrices differing in the multiplicities of the characteristic numbers, in the number and in the sizes of the blocks are there among the matrices of

?
(a)

the second order

(b)

the third order

(c)

the fourth order?

Problem VI.7.2

Find the Jordan form of the matrix and the change-of-basis matrix to the Jordan basis for the transformation given in the initial basis by the matrix

?
(a)

∥3−10−1110−1002−10010∥\left\| \begin{array}{rrrr}3 & -1 & 0 & -1 \\ 1 & 1 & 0 & -1 \\ 0 & 0 & 2 & -1 \\ 0 & 0 & 1 & 0\end{array}\right\|

(b)

∣1−10−1011101/20111000011/200001∣\left|\begin{array}{rrrrc}1 & -1 & 0 & -1 & 0 \\ 1 & 1 & 1 & 0 & 1 / 2 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 / 2 \\ 0 & 0 & 0 & 0 & 1\end{array}\right|.