**§ ****6 ****Rowdang canonical form of square matrix**

1. Invariant subspace

Let ** L** be a linear transformation of a real (or complex) linear space

Let be an invariant subspace of a linear transformation *L** of an n* -dimensional linear space *V , and **V* can use their direct sum:_{}

_{}

The necessary and sufficient condition to represent is: the matrix *A* corresponding to the linear transformation ** L** can be reduced to a block diagonal matrix under a certain basis

_{}

The order in the formula is equal to the dimension of ._{}_{}_{}

Second, the standardization of the square matrix

The form of [ Ruodang block and Ruodang standard square matrix ] is

_{}

The *m* -order square matrix of is called the Rodang block, where is an eigenvalue ._{}

The sub-matrix of a block matrix of a square matrix on the main diagonal is a block, and the rest of the sub-matrix are zero matrices, that is,

_{} ( 1 )

Then it is called the Rowdang standard square matrix or the Rowdang standard form . Note that these are not necessarily different in different blocks ._{}

[ Standardization of Square Matrix ]

1 ^{o In} the case of different eigenvalues If the eigenvalues of a square matrix *A* are not equal, then *A* can be transformed into a diagonal matrix,

The elements on its main diagonal are these eigenvalues:

_{}

2 ^{o} When the eigenvalues are equal, any square matrix *A* can be transformed into its similar Rowdang standard form ( 1 ), where _{}

is its eigenvalue and is the multiplicity of eigenvalues . If the order of the blocks is ignored, the canonical form of A is unique ._{}_{}_{}

It can be reduced to a diagonal matrix if and only if the order of all the blocks is equal to 1. This is the case of 1 ^{o} ._{}^{}

As explained above, assuming that *A* is a square matrix, then a non-singular square matrix *T* can always be found , making the square matrix similar to *A.*_{}

_{}

3. Methods and steps of square matrix standardization

[ *λ* Matrix ] Assuming that the elements of an *n* -order square matrix *A* are all complex coefficient polynomials of variable *λ* , it is called a *λ* matrix . The highest order *r* of a sub-form of a *λ* matrix that is not equal to zero is called the rank .

_{}

_{}_{}_{}

[ Invariant factor and elementary factor ] Let *r* be the rank, *k* is a positive integer , and the highest common factor of all *k* -order sub-formulas of is a polynomial, and the specified coefficient of the highest-order term is 1 ; in addition, it is specified that_{}_{}_{}_{}_{}_{}_{}_{}

_{}

say

_{}

invariant factor for ._{}

Decompose each into a first-order factor, we get_{}

_{}

Some of the exponents in the formula may be zero, and at that time , it is called an elementary factor of .
_{}_{}_{}_{}

[ Elementary Transformation · Matrix Equivalence ] A finite combination of the following three transformations on the *λ* matrix is called an elementary transformation ._{}_{}

( i ) any two rows (columns) are interchanged;

( ii ) Multiply each element of any row (column) by the same *λ* polynomial and add it to the corresponding element of another row (column);

( iii ) Multiply any row (column) element by the same complex number not equal to zero .

It should be noted that, properly implementing ( ii ), ( iii ) both transformations can yield ( i ) .

If it can be obtained by a finite number of elementary transformations, it is said to be equivalent to ._{}_{}_{}_{}_{}

*After the λ* matrix undergoes elementary transformation, its invariant factor and elementary factor remain unchanged .

[ Standard form of *λ* matrix ] Let the rank of the *λ matrix be **r* and the invariant factor be , then _{}_{}

_{}

Call the square matrix on the right the canonical form . It is determined by unique ._{}_{}

Equivalent *λ* matrices have the same canonical form .

[ eigenmatrix ] The eigenmatrix of square matrix *A* is a special *λ* matrix . So_{}

1 ^{o} If the elementary factor is_{}

_{}

which are not necessarily different from each other, then_{}

_{}

and have

_{}

2 ^{o} if *the nth* order *λ* matrix

_{}

Among them , then_{}

_{}

where *J* is the standard form of *A.*

3 ^{o} If the elementary factor of the characteristic matrix of *A is*

_{}

but
_{}

*If J* is the standard form of *A.*

[ Steps of Square Matrix Standardization ] The steps to convert the square matrix *A* into the standard form of *A* are as follows:

(1) Use elementary transformation to convert it into a diagonal matrix, decompose the polynomial on the diagonal, and get all the elementary factors .
_{}_{}

(2) Corresponding to each elementary factor , make a *m* -order Row-Dang block_{}

_{}

(3) Combine all the Ruodang blocks to get the Ruodang canonical form of *A.*

Example 1 to find a square matrix

_{}

The Jodang standard form of .

untie

_{}

It is easy to find that its invariant factors are 1 , 1 , , so the elementary factors *are*_{}_{}

_{}

Example 2 Find a square matrix

_{}

The Jodang standard form of .

untie

_{}

After elementary transformation, it can be transformed into a diagonal matrix of the following form

_{}

So the elementary factor is , , and the corresponding if-dang block is_{}_{}

_{}

Therefore , the standard form of *A 's jodang is*

_{}

Contribute a better translation