Elementary matrix

Elementary matrix

In mathematics, an elementary matrix is a simple matrix which differs from the identity matrix in a minimal way. The elementary matrices generate the general linear group of invertible matrices, and left (respectively, right) multiplication by an elementary matrix represent elementary row operations (respectively, elementary column operations).

In algebraic K-theory, "elementary matrices" refers "only" to the row-addition matrices.

Use in solving systems of equations

Elementary "row" operations do not change the solution set of the system of linear equations represented by a matrix, and are used in Gaussian elimination (respectively, Gauss-Jordan elimination) to reduce a matrix to row echelon form (respectively, reduced row echelon form).

The acronym "ero" is commonly used for "elementary row operations".

Elementary row operations do not change the kernel of a matrix (and hence do not change the solution set), but they "do" change the
image. Dually, elementary "column" operations do not change the "image", but they do change the "kernel".

There are three types of Elementary matrices (they are nxn)1) Permutation Matrix2) Diagonal Matrix3) Unipotent Matrix

Operations

There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):

;Row switching: A row within the matrix can be switched with another row.: R_i leftrightarrow R_j

;Row multiplication: Each element in a row can be multiplied by a non-zero constant.: kR_i ightarrow R_i, mbox{where } k eq 0

;Row addition: A row can be replaced by the sum of that row and a multiple of another row.: R_i + kR_j ightarrow R_i

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 obtained by swapping row "i" and row "j" of the identity matrix.

:T_{i,j} = egin{bmatrix} 1 & & & & & & & \ & ddots & & & & & & \ & & 0 & & 1 & & \ & & & ddots & & & & \ & & 1 & & 0 & & \ & & & & & & ddots & \ & & & & & & & 1end{bmatrix}quad :That is, "Tij" is the matrix produced by exchanging row "i" and row "j" of the identity matrix.

Properties

:*The inverse of this matrix is itself: "Tij−1=Tij".:*Since the determinant of the identity matrix is unity, det ["T""ij"] = −1. It follows that for any conformable square matrix "A": det ["T""ij""A"] = −det ["A"] .

Row-multiplying transformations

This transformation, "Ti"("m"), multiplies all elements on row "i" by "m" where "m" is non zero. The matrix resulting in this transformation is obtained by multiplying all elements of row "i" of the identity matrix by "m".

:T_i(m) = egin{bmatrix} 1 & & & & & & \ & ddots & & & & & \ & & 1 & & & & \ & & & m & & & \ & & & & 1 & & \ & & & & & ddots & \ & & & & & & 1end{bmatrix}quad

Properties

:*The inverse of this matrix is: "Ti"("m")−1 = "Ti"(1/"m").:*The matrix and its inverse are diagonal matrices.:*det ["T""i"(m)] = "m". Therefore for a conformable square matrix "A": det ["T""i"("m")"A"] = "m" det ["A"] .

Row-addition transformations

This transformation, "Tij"("m"), subtracts row "j" multiplied by "m" from row "i". The matrix resulting in this transformation is obtained by taking row "j" of the identity matrix, and subtracting from it "m" times row "i".:T_{i,j}(m) = egin{bmatrix} 1 & & & & & & & \ & ddots & & & & & & \ & & 1 & & & & & \ & & & ddots & & & & \ & & -m & & 1 & & \ & & & & & & ddots & \ & & & & & & & 1end{bmatrix}These are also called shear mappings or transvections.

Properties

:*"Tij"("m")−1 = "Tij"(−"m") (inverse matrix).:*The matrix and its inverse are triangular matrices.:*det ["Tij"("m")] = 1. Therefore, for a conformable square matrix "A": det ["T""ij"("1")"A"] = det ["A"] .:Row-addition transforms satisfy the Steinberg relations.

ee also

*Gaussian elimination
*Linear algebra
*System of linear equations
*Matrix (mathematics)


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