A Frobenius matrix is a special kind of square matrix from numerical mathematics. A matrix is a Frobenius matrix if it has the following three properties:
all entries on the main diagonal 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.
\( A={\begin{pmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&a_{{32}}&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&a_{{n2}}&0&\cdots &1\end{pmatrix}} \)
Frobenius matrices are invertible. 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:
\( A^{{-1}}={\begin{pmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&-a_{{32}}&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&-a_{{n2}}&0&\cdots &1\end{pmatrix}} \)
Frobenius matrices are named after Ferdinand Georg Frobenius.
The term Frobenius matrix may also be used for an alternative matrix form that differs from an Identity matrix 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.
\( {\displaystyle A={\begin{pmatrix}1&0&0&0\\0&1&0&0\\a_{31}&a_{32}&1&0\\0&0&0&1\end{pmatrix}}} \)
An alternative name for this latter form of Frobenius matrices is Gauss transformation matrix, after Carl Friedrich Gauss.[1] They are used in the process of Gaussian elimination to represent the Gaussian transformations.
If a matrix is multiplied from the left (left multiplied) with a Gauss transformation matrix, a linear combination 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).
See also
Elementary matrix, a special case of a Frobenius matrix with only one off-diagonal nonzero
Notes
Golub and Van Loan, p. 95.
References
Gene H. Golub and Charles F. Van Loan (1996). Matrix Computations, third edition, Johns Hopkins University Press. ISBN 0-8018-5413-X (hardback), ISBN 0-8018-5414-8 (paperback).
Matrix classes
Explicitly constrained entries
Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant
Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors
Companion Convergent Defective Definite Diagonalizable Hurwitz Positive-definite Stieltjes
Satisfying conditions on products or inverses
Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications
Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics
Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory
Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering
Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms
Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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