What is the theory behind elementary matrices?

In summary, elementary matrices are square matrices used in linear algebra to represent elementary row operations. They are denoted as E and are used to efficiently perform row operations on larger matrices. In solving systems of equations, elementary matrices are used to manipulate the augmented matrix and make the process more efficient. The inverse of an elementary matrix is easily calculated by reversing the row operation used to create it. Elementary matrices have various applications in linear algebra and are also useful in fields such as computer graphics and cryptography. They relate to the concept of matrix equivalence, as any matrix can be transformed into a simpler form through a series of elementary row operations.
  • #1
stanley.st
31
0
Hello,

I need to find some theory about elementary matrices. That are the matrices in the form

[tex]\mathbf{E}(\sigma,\mathbf{u},\mathbf{v})=\mathbf{I}-\frac{1}{\sigma}\mathbf{uv}^{T}[/tex]

I can't find anywhere some theory about it. Can you give me some useful links?

Thank you so much...
 
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  • #2
Hello stanley.st! :smile:

If you can't find a link, where did you come across this?

(it's not something I've seen before)
 
  • #3
Any decent linear algebra book covers elementary matrices.
 

1. What are elementary matrices?

Elementary matrices are square matrices that are used in linear algebra to represent elementary row operations. These operations include multiplying a row by a constant, swapping two rows, and adding a multiple of one row to another row. Elementary matrices are often denoted as E, and can be used to efficiently perform row operations on larger matrices.

2. How are elementary matrices used in solving systems of equations?

Elementary matrices are used in solving systems of equations by performing row operations on the augmented matrix of the system. These row operations do not change the solution set of the system, but rather manipulate the equations to make them easier to solve. By using elementary matrices, the process of solving systems of equations can be made more efficient and streamlined.

3. What is the inverse of an elementary matrix?

The inverse of an elementary matrix can be easily calculated, as it simply involves reversing the row operation that was used to create the matrix. For example, the inverse of an elementary matrix representing a row swap would simply be another elementary matrix representing the same swap. The inverse of an elementary matrix is always an elementary matrix of the same type.

4. Can elementary matrices be used for more than just solving systems of equations?

Yes, elementary matrices have many other applications in linear algebra, such as finding determinants, calculating matrix ranks, and performing matrix decompositions. They are also useful in computer graphics, cryptography, and other fields that involve matrix operations.

5. How do elementary matrices relate to the concept of matrix equivalence?

Two matrices are considered equivalent if one can be transformed into the other through a series of elementary row operations. This means that any matrix can be transformed into a simpler form through the use of elementary matrices. For example, this can be useful in finding the reduced row echelon form of a matrix or determining if a system of equations has a unique solution.

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