In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a matrix of ones is a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 where every element is equal to one. Examples of standard notation are given below:


In special contexts, the term unit matrix is used as a synonym for "matrix of ones" This is done whenever it is clear that "unit matrix" does not refer to the identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

.

Properties

For an n×n matrix of ones U, the following properties hold:

• The trace
  Trace (linear algebra)
  In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
  
   of U is n, and the determinant
  Determinant
  In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
  
   is 1 if n is 1, or 0 otherwise.
• The rank of U is 1 and the eigenvalues are n (once) and 0 (n-1 times).
• 
• The matrix is idempotent. This is a simple corollary of the above.
• where exp(U) is the matrix exponential
  Matrix exponential
  In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
  
  .
• Multiplication by U with the Hadamard product is the identity operator.