Reference

-matrix is a great tool to solve these problems such as the calculate of jordan type,the problem of ,combine with the NB watershed which was detailed explanation in the last chapter,and the last one is calculate

Some common sense.

1.-matrix is reversibledeterminant type not equls 0,the whole rank-matrix not must be reversible.
2.two -matrix are equibalentthey have same determinant factor,invariant factor,and elementary factor.
3.two matrix are similar equals they have same determinant,invariant,and elementary factor.

The core issues of -matrix

Theorem:if matrixis a n-order jordan block,other wordsthen,the minimum polynominal and elementary polynominal equals the characteristic polynominal

The solving method of jordan type

Please calcualte the jordan type of matrix,which

Theorem example:
has no repeat root,

Proposition(most important one)we know that,~
Tips: markedandwe knowbutthen

if are different any two of them,and
,

Example 1.ifis a power-zero matrix,the power-zero index is ,other words,,proof:is similar to
Tips: fromthe minimum polynominal is~,then~and~,we know that~(use the propostion)

Example 2.if,calculate the jordan type of
use the proposition,it's easy to know ~

Proposition
the minimum polynominalequals it's highest times invariant foctor.
,the minimum polynominal ofis
which is the least number who makes ,then have same minimum polynominal.

Theorem.is a n-order number matrix,is a jordan type of,for all the characteristic value,the element indiagonal is's jordan block number is

Example 1.calculate the jordan type of,which

Example 2.a n-order matrixwhen ,the jordan type ofis ,when ,the jordan type ofis

The problem of

Proposition 1
On the complex field,if,thenhave square root.
Tips: use the knowledge of~

Proposition 2
on the complex field,any reversible matrixhas quare root.

Proposition 3
on the complex field,any reversible matrixexistssuch thatfor any positive integer .

Example 1.proof:don't exists square root.

Example 2.is a n-order zero-power matrix on complex field,the index of zero-power is n,proof:doesn't have square root.

Proposition 4
a reversible matrixhas square root if and only ifhas a jordan type,and the element of the main digonal is zero's jordan block have to paired appearence byor,and only one-order jordan blockcan appear without paired.

Example 3.,proof the matrix eqution has no solution,which is a unknown 3-order complex-matrix

Example 4.is a zero-power matrix,and,proof:not exists a n-order matrixsuch that .

Example 5.is a n-order zerp-power matrix,and exists a n-order matrixsuch that,calculate the jordan type of