Changes

*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.

*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 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.

where <math>\delta_{ij}</math> is the Kronecker delta symbol.

*For example, the 5×5 shift matrices are:

*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.

*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.