writer
6,694
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − + + + + + + + + + + + + + +
no edit summary
<div style="font-size:30px">'''SHIFT'''</div><br/>
<div style="font-size:30px">'''MATRIX("SHIFT",order)'''</div><br/>
*<math>order</math> 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 ZA, ZTA, AZ, AZT, ZAZT 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 (i,j):th component of U and L are
U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1},where \delta_{ij} 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.