Difference between revisions of "Manuals/calci/SHIFT"

Revision as of 10:56, 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 $(i,j)^{th}$ 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.

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 *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 <math>(i,j)^th</math> component of U and L are:
+*The <math>(i,j)^{th}</math> component of U and L are:
-  <math>U_{ij} = \delta_{i+1,j},  L_{ij} = \delta_{i,j+1}</math>.
+  <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.