Matrix of ones
From Infogalactic: the planetary knowledge core
In mathematics, a matrix of ones or all-ones matrix is a matrix where every element is equal to one.[1] Examples of standard notation are given below:
Some sources call the all-ones matrix the unit matrix,[2] but that term may also refer to the identity matrix, a different matrix.
Properties
For an n×n matrix of ones J, the following properties hold:
- The trace of J is n,[3] and the determinant is 1 if n is 1, or 0 otherwise.
- The rank of J is 1 and the eigenvalues are n (once) and 0 (n-1 times).[4]
- J is positive semi-definite matrix. This follows from the previous property.
- [5]
- The matrix is idempotent. This is a simple corollary of the above.[5]
- where exp(J) is the matrix exponential.
- J is the neutral element of the Hadamard product.[6]
- If A is the adjacency matrix of a n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ = JA.[7]
References
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- ↑ Weisstein, Eric W., "Unit Matrix", MathWorld.
- ↑ Lua error in package.lua at line 80: module 'strict' not found..
- ↑ Stanley (2013); Horn & Johnson (2012), p. 65.
- ↑ 5.0 5.1 Lua error in package.lua at line 80: module 'strict' not found..
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