Hermite conical form

Definition.

Let be a square matrix. is said to be on Hermite canonical form (HCF) if
i) is upper triangle matrix
i) each dioganal entry is either 0 or 1
iii) the ith row of is a zero if
iv) the ith column of is if

Theorem.

Let be a matrix in HCF then is the number of 1 on the the diagonal of Also if the 1s ones are at the th positions then the th columns of form a basis of the

Proof.
The no. of zero rows in is so, row rank of But the columns are which are linearly independent , so thre are atleast linearly independent vectors hence

forms a basis of

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let be a group of all bijections on the set
be a set of all bijections from we can define a binary operation on as follows

this makes as a group

let be a permutation. Then there is a matrix associated to say st. the ith row of is where ie the ith row of is
let

A permutation matrix is a square matrix, containing only 's and 1's more over excatly one 1 in each row and one in each column

Every permutation matrix is for some

Exercise.

det

Pre-multiplication of a matrix by a permutation matrix permutes the rows of A, the ith row of it the the row of A
the post multiplication of permutes the columns of A th column of A permutes to of A

Lemma.

Let be a matrix in HCF with rank then there exist a permutaion matrix such that is of the form

Proposition (If

is a HCF then

is an idempotent).

Proof.

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Exercise.

If is a permutaion matrix what is where

#17-oct

Theorem.

Any square matrix can be reduced to a matrix in HCF by elementary row operations

Proof.
Let A be a matrix
we say that the first columns of A have Hermite property is they are the first columns of a matrix in the HCF, ie

i)
ii)
iii)
iv)

If the first column is a zero column then it is already in Hermite form.
If first column is non-zero, and then we normalize to 1 by dividing row 1 by if then we do a row interchange and make and repeat as in the case
we make the entries by elementary row operations this makes the first column in hermite for suppose by elementary row operations we made the first columns into hermite form.

Case 1:
There exist a non-zero entry on or below the diagonal in the th column,ie an
such that if is not zero then do a suitable row interchange normalize into a pivot and make all other entries in the column zeros. if then normalize and proceed

Case 2:

If the jth column is zero then we are done

case 3:

and ie all elements on and below the diagonal in the jth column are zeros but a non-zero element but
then we interchange the row and make all entries in the colum zeros using as the pivot

Case 4:
and for either or we leave it as it is

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Let be a system of equation where A is form the matrix

If A is non-singular applying row operation of A reduces A to its RREF
and applying the same operation on gives us
applying same operations on gives us the unique solution

If A is singular reduce A to its HCF by applying row operations and apply same row operation on and as well Let us arrive at the following

Remark.

let be a particular solution of all general solutions of is

Theorem.

The system is consistent iff whenever then is a particular solution and is a general solution for any arbitrary
The non-zero columns of form a basis of the is a G-Inverse G-Inverse of a if the in the dioganal of occur in th places BC as a rank factorization of A where
and where is the ith column of A and is the row of

Proof.

Let be the product of all the row operations applied

As is Hermite form is idempotent.
this shows is the g-inv of A
is an idempotent

is a g-inv of so,
so is the solution of then is a solution of is

Now But is idempotent so so a general solution of is ie, for any

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Let be a linear map let be a basis of
Matrix of with respective to say A is obtained as follows ith column of A is the coordinate vector of wrt

Let
an invertable matrix such that

Claim.

The matrix of wrt new basis in