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::

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

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:$p\_U(lambda)\; =\; (-1)^nlambda^n.$
* "U"^"n" = 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-a""U"^"n-a" + "L"^"a""U"^"a" = "U"^"n-a""L"^"n-a" + "U"^"a""L"^"a" = "I" (the identity matrix), for any integer "a" between 0 and "n" inclusive.

Examples

::

Then

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.

:::::

ee also

* Nilpotent matrix

References