Sparseness for a matrix I would like to define a function $f$ whose range is $[0,1]$ such that it takes a matrix $C \in R_+$ of dimension $m \times n$. The entries in the matrices are also in the range $[0,1]$. In addition, each row of $C$ sums to $1$.
The function $f$ should produce $0$ when all the entries in the matrix $C$ are same and produce $1$ when there is only one $1$ entry in each row.
For example,when  
C = [1 0 0 0
     0 0 1 0
     0 0 0 1
     0 1 0 0]

$f(C) = 1$
In other case, when  
C = [0.25 0.25 0.25 0.25
     0.25 0.25 0.25 0.25
     0.25 0.25 0.25 0.25
     0.25 0.25 0.25 0.25]

$f(C) = 0$
Can anyone help me out in contructing such a function?
 A: Let $R_i$ be the maximum value in row $i$ less the minimum value in that row, and put $f = (R_1 + ..R_n)/n$.
A: Let the matrix have the form
$$
\begin{pmatrix}
  p_{1,1} & \ldots & p_{1,n}  \\
  \vdots & \ddots & \vdots  \\
  p_{n,1} & \ldots & p_{n,n}
\end{pmatrix}
.
$$
I would approach this problem in the following way:
Since $\sum_{i=1}^np_{k,i}=1$ for all $k$, each of the matrix rows can be thought of as a probability distribution.
The almost canonical characterisation of a probability distribution is its Shannon entropy
$$
  H_k = -\sum_ip_{k,i}\cdot\ln(p_{k,i})
$$
where summands with $p_{k,i}=0$ are left out.
This is known to be highest when $p_{k,i}=p_{k,j}\equiv p_k=\tfrac1n$, and lowest when $p_{k,i}=0$ save for one $i$. Namely,
$$
  \bigl.H_k\bigr|_{p_{k,i}=p_{k,j}}
  = -\sum_ip_{k}\cdot\ln(p_{k})
  = -n\cdot p_{k}\cdot\ln(p_{k})
  = -n\cdot \tfrac1n\cdot\ln(\tfrac1n)
  = -\ln(\tfrac1n)
  = \ln(n)
$$
and
$$
  \bigl.H_k\bigr|_{p_{k,i}=0\ \forall i\neq j}
  = -0 - \ldots - p_{k,j}\ln (p_{k,j}) - \ldots -0
  = -1\cdot \ln(1)
  = 0
$$
We can now define a "total entropy" of the matrix,
$$
  H := -\sum_{k}\sum_{i}p_{k,i}\cdot\ln(p_{k,i})
$$
This will now be $n\ln(n)$ for the matrix with all-equal entries, and $0$ for a matrix with only-$0$-and-$1$-entries. To get a value in your desired range, just do an affine transformation
$$
  f:= 1-\frac{H}{n\ln(n)}
  = 1 + \sum_{k,i}\frac{p_{k,i}\ln(p_{k,i})}{n\cdot\ln(n)}
$$
