Can Any Matrix be Reduced to a Canonical Matrix?

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In summary, the conversation discusses whether or not any matrix can be reduced to a canonical matrix using the three elementary operations. The speaker believes that the answer is yes and explains that the only way a linear system would have no solution is if the lowest non-zero row has a specific form. They also mention that the desired form is reduced row echelon form. The other speaker asks for clarification on what is meant by a canonical matrix and suggests using row echelon form to justify the answer. The original speaker clarifies that it was not a homework question, just a yes/no question to check their understanding.
  • #1
daniel_i_l
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Homework Statement


I have a question that doesn't relate to a specific problem:
Can any matrix be reduced to a canonical matrix?

Homework Equations



the three elementry operations

The Attempt at a Solution



I think that the answer is yes and that the only way that a linear system has no answer is when the lowest non-zero row has the form of: (0,..,0,1)
Is that right?
Thanks.
 
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  • #2
What is a 'canonical' matrix? Jordan canonical form? Row echelon form? Reduced row echelon form?
 
  • #3
Reduced row echelon form.
 
  • #4
Have you tried to put matrices in row ecehlon for using the operations? It's always worked, right. NOw, do you need to justify that, or is it just a yes/no question?
 
  • #5
This wasn't a homework question, i just was asking to see if i understood correctly, so it's a yes/no question.
 

FAQ: Can Any Matrix be Reduced to a Canonical Matrix?

What is a canonical matrix?

A canonical matrix is a matrix that has been transformed into a specific, standard form through a series of operations such as row operations and scaling. The resulting matrix has certain properties that make it useful for solving systems of equations and other mathematical problems.

Can any matrix be reduced to a canonical matrix?

Yes, with the use of row operations and scaling, any matrix can be reduced to a canonical form. However, the resulting canonical matrix may not be unique as there are different types of canonical forms.

Why is it useful to reduce a matrix to a canonical form?

Reducing a matrix to a canonical form can make it easier to solve systems of equations, perform matrix operations, and analyze the properties of the matrix. It also allows for easier comparison and classification of matrices.

What are the common types of canonical forms?

Two common types of canonical forms are the row-echelon form and the reduced row-echelon form. The row-echelon form has leading 1's in each row, while the reduced row-echelon form has leading 1's in each row and column, and all other elements are reduced to 0.

Can a matrix be reduced to more than one type of canonical form?

Yes, a matrix can be reduced to multiple types of canonical forms depending on the operations used. For example, the same matrix can be reduced to both the row-echelon form and the reduced row-echelon form.

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