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Elementary matrix transformations
Elementary matrix transformations or Elementary row and column transformations are linear transformations which are normally used in gauss elimination to solve a set of linear equations.
We distinguish three types of elementary transformations and their corresponding matrices:
- Row switching transformations,
- Row multiplying transformations,
- Linear combinator transformations.
1. Row switching transformations
This transformation, Tij, switches all matrix elements on row i with their counterparts on row j. The matrix resulting in this transformation is:
Properties
- *The inverse of this matrix is itself: Tij-1=Tij.
- *When applied to a matrix A: det[TA]=-det[A].
- *The matrix and it's inverse are lower triangular matrices.
2. Row multiplying transformations
This transformation, Ti(m), multiplies all elements on row i with m. The matrix resulting in this transformation is:
Properties
- *The inverse of this matrix is: Ti(m)-1=Ti(1/m).
- *When applied to a matrix A: det[TA]=mdet[A].
- *The matrix and it's inverse are lower triangular matrices.
3. Linear combinator transformations
This transformation, Tij(m), substracts row i multiplied by m from row j. The matrix resulting in this transformation is:
Properties
- *The inverse of this matrix is: Tij(m)-1=Tij(-m).
- *When applied to a matrix A: det[TA]=det[A].
- *The matrix and it's inverse are lower triangular matrices.
See also
Referenced By
Doolittle decomposition | LU-factorization | LU decomposition | LU factorization | List of mathematical topics (D-F) | List of mathematical topics (F-Z)
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