Shift matrix

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_{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

U_5=\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} \quad L_5=\begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix}.

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.^[1]

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(\lambda) = (-1)^n\lambda^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:

L^T = U; U^T = L

The null spaces of U and L are

N(U) = \operatorname{span}\{ (1,0,\ldots, 0)^T \},

N(L) = \operatorname{span}\{ (0,\ldots, 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 $(1,0,\ldots, 0)^T$ , and the only eigenvector for L is $(0,\ldots, 0,1)^T$ .

For LU and UL we have

UL = I - \operatorname{diag}(0,\ldots, 0,1),

LU = I - \operatorname{diag}(1,0,\ldots, 0).

These matrices are both idempotent, symmetric, and have the same rank as U and L

L^n-aU^n-a + L^aU^a = U^n-aL^n-a + U^aL^a = 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

\begin{pmatrix} S_1 & 0 & \ldots & 0 \\ 0 & S_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & S_r \end{pmatrix}

where each of the blocks S₁, S₂, ..., S_r is a shift matrix (possibly of different sizes).^[2]^[3]

Examples

S=\begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix}; \quad A=\begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 1 \\ 1 & 2 & 3 & 2 & 1 \\ 1 & 2 & 2 & 2 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{pmatrix}.

Then $SA=\begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 1 \\ 1 & 2 & 3 & 2 & 1 \\ 1 & 2 & 2 & 2 & 1 \end{pmatrix}; \quad AS=\begin{pmatrix} 1 & 1 & 1 & 1 & 0 \\ 2 & 2 & 2 & 1 & 0 \\ 2 & 3 & 2 & 1 & 0 \\ 2 & 2 & 2 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 \end{pmatrix}.$

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

S^{T}AS=\begin{pmatrix} 2 & 2 & 2 & 1 & 0 \\ 2 & 3 & 2 & 1 & 0 \\ 2 & 2 & 2 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}.

Notes

↑ Beauregard & Fraleigh (1973, p. 312)
↑ Beauregard & Fraleigh (1973, pp. 312,313)
↑ Herstein (1964, p. 250)

References

Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X
Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016

External links

Shift Matrix - entry in the Matrix Reference Manual

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