Linear transformations on symmetric matrices II

Let Tbe a linear mapping on the space of n× nsymmetric matrices over a field Fof characteristic not equal to two. We obtain the structure of Tfor the following cases:(i) Tpreserves matrices of rank less than three; (ii) Tpreserves nonzero matrices of rank less than K + 1 where Kis a fixed positive integer less than nand Fis algebraically closed; (iii) Tpreserves rank Kmatrices where Kis a fixed odd integer and Fis algebraically closed.

[1]  M. Marcus,et al.  Linear Transformations on Algebras of Matrices , 1959, Canadian Journal of Mathematics.

[2]  Marvin Marcus,et al.  Transformations on tensor product spaces , 1959 .

[3]  M. Marcus,et al.  Linear maps on skew symmetric matrices: the invariance of elementary symmetric functions , 1960 .

[4]  J. Adams.  Vector fields on spheres , 1962 .

[5]  H. Flanders.  On Spaces of Linear Transformations with Bounded Rank , 1962 .

[6]  Decompositions of $E^{3}$ which yield $E^{3}$. , 1967 .

[7]  R. Westwick,et al.  Transformations on tensor spaces. , 1967 .

[8]  D. Djoković,et al.  Linear transformations of tensor products preserving a fixed rank. , 1969 .

[9]  Decomposable symmetric tensors. , 1970 .

[10]  LeRoy B. Beasley.  Linear transformations on matrices: the invariance of rank k matrices , 1970 .

[11]  M. Marcus.  Linear transformations on matrices , 1971 .

[12]  R. Westwick.  Spaces of linear transformations of equal rank , 1972 .

[13]  L. Cummings.  TRANSFORMATIONS OF SYMMETRIC TENSORS , 1972 .

[14]  M. Marcus.  Finite dimensional multilinear algebra , 1973 .

[15]  L. Carlitz,et al.  Enumeration of matrices with prescribed row and column sums , 1973 .

[16]  Linear Transformations on Grassmann Spaces III , 1974 .

[17]  Linear transformations on symmetric spaces. , 1974 .

[18]  Ronald E. Kutz.  Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups , 1974 .

[19]  A Note on Maximal Decomposable Subspaces of Symmetric Spaces , 1975 .

[20]  J. Humphreys,et al.  Linear Algebraic Groups , 1975 .

[21]  Linear Operators Preserving The Real Symplectic Group , 1975, Canadian Journal of Mathematics.

[22]  H. Minc.  The invariance of elementary symmetric functions , 1976 .

[23]  M. Lim.  Linear transformation on symmetry classes of tensors , 1976 .

[24]  Peter Botta.  Linear maps that preserve singular and nonsingular matrices , 1978 .

[25]  Linear transformations on symmetric matrices that preserve the permanent , 1978 .