Manuals/calci/SHIFT

From ZCubes Wiki
Jump to navigation Jump to search
MATRIX("SHIFT",order)


  • is the size of the Shift matrix.

Description

  • This function returns shift matrix of order 3.
  • A shift matrix is a binary matrix with ones only on the superdiagonal or subdiagonal, and zeroes elsewhere.
  • A shift matrix U with ones on the superdiagonal is an upper shift matrix.
  • The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix.
  • Let Z is a shift matrix , then are equal to the matrix A shifted one position down, up left, right, and down along the main diagonal respectively.
  • The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix.
  • The component of U and L are:
.

where is the Kronecker delta symbol.

  • For example, the 5×5 shift matrices are

  • All shift matrices are nilpotent; an n by n shift matrix S becomes the null matrix when raised to the power of its dimension n.


Examples

  • 1.MATRIX("shift") = 0
  • 2.MATRIX("shift",3)
0 1 0
0 0 1
0 0 0
  • 3.MATRIX("shift",7)
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 0 0 0 0

See Also

References