In mathematics, a Redheffer matrix, often denoted A n {\displaystyle A_{n}} as studied by Redheffer (1977), is a square (0,1) matrix whose entries aij are 1 if i divides j or if j = 1; otherwise, aij = 0. It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix products involving the transpose of the n t h {\displaystyle n^{th}} Redheffer matrix.
Variants and definitions of component matrices Since the invertibility of the Redheffer matrices are complicated by the initial column of ones in the matrix, it is often convenient to express A n := C n + D n {\displaystyle A_{n}:=C_{n}+D_{n}} where C n := [ c i j ] {\displaystyle C_{n}:=[c_{ij}]} is defined to be the (0,1) matrix whose entries are one if and only if j = 1 {\displaystyle j=1} and i ≠ 1 {\displaystyle i\neq 1} . The remaining one-valued entries in A n {\displaystyle A_{n}} then correspond to the divisibility condition reflected by the matrix D n {\displaystyle D_{n}} , which plainly can be seen by an application of Mobius inversion is always invertible with inverse D n − 1 = [ μ ( j / i ) M i ( j ) ] {\displaystyle D_{n}^{-1}=\left[\mu (j/i)M_{i}(j)\right]} . We then have a characterization of the singularity of A n {\displaystyle A_{n}} expressed by det ( A n ) = det ( D n − 1 C n + I n ) . {\displaystyle \det \left(A_{n}\right)=\det \left(D_{n}^{-1}C_{n}+I_{n}\right).}
If we define the function
M j ( i ) := { 1 , if j divides i; 0 , otherwise, , {\displaystyle M_{j}(i):={\begin{cases}1,&{\text{ if j divides i; }}\\0,&{\text{otherwise, }}\end{cases}},}
then we can define the n t h {\displaystyle n^{th}} Redheffer (transpose) matrix to be the nxn square matrix R n = [ M j ( i ) ] 1 ≤ i , j ≤ n {\displaystyle R_{n}=[M_{j}(i)]_{1\leq i,j\leq n}} in usual matrix notation. We will continue to make use this notation throughout the next sections.
Examples The matrix below is the 12 × 12 Redheffer matrix. In the split sum-of-matrices notation for A 12 := C 12 + D 12 {\displaystyle A_{12}:=C_{12}+D_{12}} , the entries below corresponding to the initial column of ones in C n {\displaystyle C_{n}} are marked in blue.
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