Difference between revisions of "Manuals/calci/SHIFT"

Revision as of 11:53, 4 May 2015

MATRIX("SHIFT",order)

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 $ZA,Z^{T}A,AZ,AZ^{T},ZAZ^{T}$ 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 $(i,j)^{t}h$ 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.

@@ Line 7: / Line 7: @@
 *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.
+*Let Z is a shift matrix , then <math>ZA, Z^TA, AZ, AZ^T, ZAZ^T</math> 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
+*The <math>(i,j)^th</math> 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.
+  <math>U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1}</math>,where <math>\delta_{ij}</math> 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.