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Definition

The centering matrix of size n is defined as the n-by-n matrix

C n = I n − 1 n J n {\displaystyle C_{n}=I_{n}-{\tfrac {1}{n}}J_{n}}

where I n {\displaystyle I_{n}\,} is the identity matrix of size n and J n {\displaystyle J_{n}} is an n-by-n matrix of all 1's.

For example

C 1 = [ 0 ] {\displaystyle C_{1}={\begin{bmatrix}0\end{bmatrix}}} , C 2 = [ 1 0 0 1 ] − 1 2 [ 1 1 1 1 ] = [ 1 2 − 1 2 − 1 2 1 2 ] {\displaystyle C_{2}=\left[{\begin{array}{rrr}1&0\\0&1\end{array}}\right]-{\frac {1}{2}}\left[{\begin{array}{rrr}1&1\\1&1\end{array}}\right]=\left[{\begin{array}{rrr}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{array}}\right]} , C 3 = [ 1 0 0 0 1 0 0 0 1 ] − 1 3 [ 1 1 1 1 1 1 1 1 1 ] = [ 2 3 − 1 3 − 1 3 − 1 3 2 3 − 1 3 − 1 3 − 1 3 2 3 ] {\displaystyle C_{3}=\left[{\begin{array}{rrr}1&0&0\\0&1&0\\0&0&1\end{array}}\right]-{\frac {1}{3}}\left[{\begin{array}{rrr}1&1&1\\1&1&1\\1&1&1\end{array}}\right]=\left[{\begin{array}{rrr}{\frac {2}{3}}&-{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {2}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&-{\frac {1}{3}}&{\frac {2}{3}}\end{array}}\right]}

Properties

Given a column-vector, v {\displaystyle \mathbf {v} \,} of size n, the centering property of C n {\displaystyle C_{n}\,} can be expressed as

C n v = v − ( 1 n J n , 1 T v ) J n , 1 {\displaystyle C_{n}\,\mathbf {v} =\mathbf {v} -({\tfrac {1}{n}}J_{n,1}^{\textrm {T}}\mathbf {v} )J_{n,1}}

where J n , 1 {\displaystyle J_{n,1}} is a column vector of ones and 1 n J n , 1 T v {\displaystyle {\tfrac {1}{n}}J_{n,1}^{\textrm {T}}\mathbf {v} } is the mean of the components of v {\displaystyle \mathbf {v} \,} .

C n {\displaystyle C_{n}\,} is symmetric positive semi-definite.

C n {\displaystyle C_{n}\,} is idempotent, so that C n k = C n {\displaystyle C_{n}^{k}=C_{n}} , for k = 1 , 2 , … {\displaystyle k=1,2,\ldots } . Once the mean has been removed, it is zero and removing it again has no effect.

C n {\displaystyle C_{n}\,} is singular. The effects of applying the transformation C n v {\displaystyle C_{n}\,\mathbf {v} } cannot be reversed.

C n {\displaystyle C_{n}\,} has the eigenvalue 1 of multiplicity n − 1 and eigenvalue 0 of multiplicity 1.

C n {\displaystyle C_{n}\,} has a nullspace of dimension 1, along the vector J n , 1 {\displaystyle J_{n,1}} .

C n {\displaystyle C_{n}\,} is an orthogonal projection matrix. That is, C n v {\displaystyle C_{n}\mathbf {v} } is a projection of v {\displaystyle \mathbf {v} \,} onto the (n − 1)-dimensional subspace that is orthogonal to the nullspace J n , 1 {\displaystyle J_{n,1}} . (This is the subspace of all n-vectors whose components sum to zero.)

The trace of C n {\displaystyle C_{n}} is n ( n − 1 ) / n = n − 1 {\displaystyle n(n-1)/n=n-1} .

Application

Although multiplication by the centering matrix is not a computationally efficient way of removing the mean from a vector, it is a convenient analytical tool. It can be used not only to remove the mean of a single vector, but also of multiple vectors stored in the rows or columns of an m-by-n matrix X {\displaystyle X} .

The left multiplication by C m {\displaystyle C_{m}} subtracts a corresponding mean value from each of the n columns, so that each column of the product C m X {\displaystyle C_{m}\,X} has a zero mean. Similarly, the multiplication by C n {\displaystyle C_{n}} on the right subtracts a corresponding mean value from each of the m rows, and each row of the product X C n {\displaystyle X\,C_{n}} has a zero mean. The multiplication on both sides creates a doubly centred matrix C m X C n {\displaystyle C_{m}\,X\,C_{n}} , whose row and column means are equal to zero.

The centering matrix provides in particular a succinct way to express the scatter matrix, S = ( X − μ J n , 1 T ) ( X − μ J n , 1 T ) T {\displaystyle S=(X-\mu J_{n,1}^{\mathrm {T} })(X-\mu J_{n,1}^{\mathrm {T} })^{\mathrm {T} }} of a data sample X {\displaystyle X\,} , where μ = 1 n X J n , 1 {\displaystyle \mu ={\tfrac {1}{n}}XJ_{n,1}} is the sample mean. The centering matrix allows us to express the scatter matrix more compactly as

S = X C n ( X C n ) T = X C n C n X T = X C n X T . {\displaystyle S=X\,C_{n}(X\,C_{n})^{\mathrm {T} }=X\,C_{n}\,C_{n}\,X\,^{\mathrm {T} }=X\,C_{n}\,X\,^{\mathrm {T} }.}

C n {\displaystyle C_{n}} is the covariance matrix of the multinomial distribution, in the special case where the parameters of that distribution are k = n {\displaystyle k=n} , and p 1 = p 2 = ⋯ = p n = 1 n {\displaystyle p_{1}=p_{2}=\cdots =p_{n}={\frac {1}{n}}} .

References

1. John I. Marden, Analyzing and Modeling Rank Data, Chapman & Hall, 1995, ISBN 0-412-99521-2, page 59. /wiki/ISBN_(identifier) ↩