Samiksha Jaiswal (Editor)

Shift matrix

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In mathematics, 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. The (i,j):th component of U and L are

U i j = δ i + 1 , j , L i j = δ i , j + 1 ,

where δ i j is the Kronecker delta symbol.

For example, the 5×5 shift matrices are

Clearly, the transpose of a lower shift matrix is an upper shift matrix and vice versa.

As a linear transformation, a lower shift matrix shifts the components of a row vector one position to the right, with a zero appearing in the first position. An upper shift matrix shifts the components of a row vector one position to the left, with a zero appearing in the last position.

Premultiplying a matrix A by a lower shift matrix results in the elements of A being shifted downward by one position, with zeroes appearing in the top row. Postmultiplication by a lower shift matrix results in a shift left. Similar operations involving an upper shift matrix result in the opposite shift.

Clearly 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.

Properties

Let L and U be the n by n lower and upper shift matrices, respectively. The following properties hold for both U and L. Let us therefore only list the properties for U:

  • det(U) = 0
  • trace(U) = 0
  • rank(U) = n−1
  • The characteristic polynomials of U is
  • p U ( λ ) = ( 1 ) n λ n .
  • Un = 0. This follows from the previous property by the Cayley–Hamilton theorem.
  • The permanent of U is 0.
  • The following properties show how U and L are related:

  • LT = U; UT = L
  • The null spaces of U and L are
  • N ( U ) = span { ( 1 , 0 , , 0 ) T } , N ( L ) = span { ( 0 , , 0 , 1 ) T } .
  • The spectrum of U and L is { 0 } . The algebraic multiplicity of 0 is n, and its geometric multiplicity is 1. From the expressions for the null spaces, it follows that (up to a scaling) the only eigenvector for U is ( 1 , 0 , , 0 ) T , and the only eigenvector for L is ( 0 , , 0 , 1 ) T .
  • For LU and UL we have
  • U L = I diag ( 0 , , 0 , 1 ) , L U = I diag ( 1 , 0 , , 0 ) . These matrices are both idempotent, symmetric, and have the same rank as U and L
  • Ln-aUn-a + LaUa = Un-aLn-a + UaLa = I (the identity matrix), for any integer a between 0 and n inclusive.
  • If N is any nilpotent matrix, then N is similar to a block diagonal matrix of the form

    ( S 1 0 0 0 S 2 0 0 0 S r )

    where each of the blocks S1S2, ..., Sr is a shift matrix (possibly of different sizes).

    Then S A = ( 0 0 0 0 0 1 1 1 1 1 1 2 2 2 1 1 2 3 2 1 1 2 2 2 1 ) ; A S = ( 1 1 1 1 0 2 2 2 1 0 2 3 2 1 0 2 2 2 1 0 1 1 1 1 0 ) .

    Clearly there are many possible permutations. For example, S T A S is equal to the matrix A shifted up and left along the main diagonal.

    References

    Shift matrix Wikipedia