Matrix of ones

In

Some sources call the all-ones matrix the unit matrix,[2] but that term may also refer to the identity matrix, a different type of matrix.

A vector of ones or all-ones vector is matrix of ones having row or column form; it should not be confused with unit vectors.

Properties

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

• The trace of J equals n,^[3] and the determinant equals 0 for n ≥ 2, but equals 1 if n = 1.
• The characteristic polynomial of J is ${\displaystyle (x-n)x^{n-1}}$.
• The minimal polynomial of J is ${\displaystyle x^{2}-nx}$.
• The
  eigenvalues are n with multiplicity 1 and 0 with multiplicity n − 1.^[4]
• ${\displaystyle J^{k}=n^{k-1}J}$ for ${\displaystyle k=1,2,\ldots .}$[5]
• J is the
  neutral element of the Hadamard product.^[6]

When J is considered as a matrix over the real numbers, the following additional properties hold:

• J is
  positive semi-definite matrix
  .
• The matrix ${\displaystyle {\tfrac {1}{n}}J}$ is idempotent.^[5]
• The matrix exponential of J is ${\displaystyle \exp(\mu J)=I+{\frac {e^{\mu n}-1}{n}}J}$

Applications

The all-ones matrix arises in the mathematical field of

See also

• Zero matrix, a matrix where all entries are zero
• Single-entry matrix

References

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