Matrix of ones - Alchetron, The Free Social Encyclopedia

In mathematics, a matrix of ones or all-ones matrix is a matrix over the real numbers where every element is equal to one. Examples of standard notation are given below:

J 2 = ( 1 1 1 1 ) ; J 3 = ( 1 1 1 1 1 1 1 1 1 ) ; J 2 , 5 = ( 1 1 1 1 1 1 1 1 1 1 ) .

Some sources call the all-ones matrix the unit matrix, 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 characteristic polynomial of J is ( x − n ) x n − 1 .

The trace of J is n, and the determinant is 1 if n is 1, or 0 otherwise.

The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity n−1.

J is positive semi-definite matrix. This follows from the previous property.

J k = n k − 1 J , for k = 1 , 2 , … .

The matrix 1 n J is idempotent. This is a simple corollary of the above.

exp ⁡ ( J ) = I + e n − 1 n J , where exp(J) is the matrix exponential.

J is the neutral element of the Hadamard product.

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.

References

Matrix of ones Wikipedia

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