A Frobenius matrix is a special kind of

square matrix In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Squ ...

from

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

. A matrix is a Frobenius matrix if it has the following three properties: * all entries on the

main diagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix ...

are ones * the entries below the main diagonal of at most one column are arbitrary * every other entry is zero The following matrix is an example. :

A=\begin
  1    &   0    &   0    & \cdots & 0 \\
  0    &   1    &   0    & \cdots & 0 \\
  0    & a_ &   1    & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
  0    & a_ &   0    & \cdots & 1
\end

Frobenius matrices are

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...

. The inverse of a Frobenius matrix is again a Frobenius matrix, equal to the original matrix with changed signs outside the main diagonal. The inverse of the example above is therefore: :

A^=\begin
  1    &    0    &   0    & \cdots & 0 \\
  0    &    1    &   0    & \cdots & 0 \\
  0    & -a_ &   1    & \cdots & 0 \\
\vdots & \vdots  & \vdots & \ddots & \vdots \\
  0    & -a_ &   0    & \cdots & 1
\end

Frobenius matrices are named after

Ferdinand Georg Frobenius Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famou ...

. The term Frobenius matrix may also be used for an alternative matrix form that differs from an

Identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...

only in the elements of a single row preceding the diagonal entry of that row (as opposed to the above definition which has the matrix differing from the identity matrix in a single column below the diagonal). The following matrix is an example of this alternative form showing a 4-by-4 matrix with its 3rd row differing from the identity matrix. :

A=\begin
  1    &   0    &   0    & 0 \\
  0    &   1    &   0    & 0 \\
a_ & a_ &   1    & 0 \\
  0    &   0    &   0    & 1
\end

An alternative name for this latter form of Frobenius matrices is Gauss transformation matrix, after

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

.Golub and Van Loan, p. 95. They are used in the process of

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...

to represent the Gaussian transformations. If a matrix is multiplied from the left (left multiplied) with a Gauss transformation matrix, a

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

of the preceding rows is added to the given row of the matrix (in the example shown above, a linear combination of rows 1 and 2 will be added to row 3). Multiplication with the inverse matrix subtracts the corresponding linear combination from the given row. This corresponds to one of the elementary operations of Gaussian elimination (besides the operation of transposing the rows and multiplying a row with a scalar multiple).

Notes

References

Gene H. Golub Gene Howard Golub (February 29, 1932 – November 16, 2007), was an American numerical analysis, numerical analyst who taught at Stanford University as Fletcher R. Jones, Fletcher Jones Professor of Computer Science and held a courtesy appointmen ...

and

Charles F. Van Loan Charles Francis Van Loan (born September 20, 1947) is an emeritus professor of computer science and the Joseph C. Ford Professor of Engineering at Cornell University,

(1996). ''Matrix Computations'', third edition, Johns Hopkins University Press. (hardback), (paperback). {{Matrix classes Matrices (mathematics)

See also

Notes

References