Real and complex canonical forms

In summary, the conversation discusses finding the canonical forms over R and C for a given quadratic form. The process involves transforming the matrix to an identity matrix or a matrix with a 0 row. The canonical form over R is diag(I_r, -I_s, O_t) and the canonical form over C is diag(I_r,0_t). The question of uniqueness of the canonical form is also raised. It is explained that canonical refers to the usual way of representing something, and that a canonical form for a quadratic form is one without mixed terms and only squares. A resource for further reading on this topic is also provided.
  • #1
alvielwj
20
0
A question about how find the canonical forms over R and C.
An example, given a quadratic form,q(x,y,z)=x^2 + 2xy + 4yz + z^2
find the canonical forms over R and C.

First step,i get the matrix 1 2^0.5 0
2^0.5 0 2
0 2 1
then by doing the double operation
i get the identiy matrix.
the canonical form over R is diag(I_r, -I_s, O_t)
and the canonical form over C is diag(I_r,0_t)
is the canonical form unique?
what are the final anwsers?

I know that any matrix can be changed to Identity matrix or a matrix with a 0 row.
does it mean most matries have similar canonical form?
 
Physics news on Phys.org
  • #2
Canonical is no property which is defined in any way. It literally means: "How it is usually done (written in the canon)". So whenever there is a usual way to the goal, then it is said canonical. In this case a canonical form, better standard form, is the one which has no mixed terms ##x_ix_j## and only squares ##x_i^2##.

See http://www.maths.qmul.ac.uk/~twm/MTH6140/la26.pdf
 

1. What is the difference between real and complex canonical forms?

The real canonical form is a matrix representation of a linear transformation, where the matrix is in diagonal form with the eigenvalues of the transformation on the main diagonal. The complex canonical form is a similar concept, but it is used for complex matrices and includes both real and imaginary eigenvalues.

2. How can real and complex canonical forms be used in linear algebra?

Real and complex canonical forms are used to simplify and understand linear transformations in linear algebra. They provide a way to represent a transformation in a diagonal form, making it easier to analyze its properties and behavior.

3. What is the significance of eigenvalues in real and complex canonical forms?

Eigenvalues are the values along the main diagonal of the canonical form matrix. They represent the scaling factor for each eigenvector in the transformation. In real canonical form, the eigenvalues are real numbers, while in complex canonical form, they can be complex numbers.

4. How are real and complex canonical forms related to diagonalization?

Diagonalization is the process of finding a diagonal matrix that is similar to a given matrix. Real and complex canonical forms are diagonal matrices, making them a special case of diagonalization. They are used to simplify the process of diagonalization for linear transformations.

5. Can real and complex canonical forms be used for non-square matrices?

No, real and complex canonical forms can only be used for square matrices. This is because the number of eigenvalues in the canonical form must match the dimension of the matrix. Non-square matrices do not have the same number of rows and columns, so they cannot be represented in a canonical form.

Similar threads

  • Linear and Abstract Algebra
Replies
4
Views
2K
Replies
7
Views
782
  • Linear and Abstract Algebra
Replies
14
Views
2K
  • Linear and Abstract Algebra
Replies
6
Views
1K
  • Linear and Abstract Algebra
Replies
11
Views
2K
  • Linear and Abstract Algebra
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
6K
  • Linear and Abstract Algebra
Replies
1
Views
863
  • Linear and Abstract Algebra
Replies
20
Views
18K
  • Special and General Relativity
Replies
8
Views
4K
Back
Top