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Inmathematics andmultivariate statistics, thecentering matrix^[1] is asymmetric andidempotent matrix, which when multiplied with a vector has the same effect as subtracting themean of the components of the vector from every component of that vector.

Thecentering matrix of sizen is defined as then-by-n matrix

${\displaystyle C_n=I_n-\frac{1}{n}{J}_n}$

where${\displaystyle I_n}$ is theidentity matrix of sizen and${\displaystyle J_n}$ is ann-by-nmatrix of all 1's.

For example

${\displaystyle C_1=\left[\begin{array}{c}0\end{array}\right]}$,

,

Given a column-vector,${\displaystyle \mathbf{v}}$ of sizen, thecentering property of${\displaystyle C_n}$ can be expressed as

${\displaystyle C_n\mathbf{v}=\mathbf{v}-(\frac{1}{n}{J}_{n,1}^{\text{T}}\mathbf{v})J_{n,1}}$

where${\displaystyle J_{n,1}}$ is acolumn vector of ones and${\displaystyle \frac{1}{n}{J}_{n,1}^{\text{T}}\mathbf{v}}$ is the mean of the components of${\displaystyle \mathbf{v}}$.

${\displaystyle C_n}$ is symmetricpositive semi-definite.

${\displaystyle C_n}$ isidempotent, so that${\displaystyle C_n^k=C_n}$, for${\displaystyle k=1,2,\dots }$. Once the mean has been removed, it is zero and removing it again has no effect.

${\displaystyle C_n}$ is singular. The effects of applying the transformation${\displaystyle C_n\mathbf{v}}$ cannot be reversed.

${\displaystyle C_n}$ has theeigenvalue 1 of multiplicityn − 1 and eigenvalue 0 of multiplicity 1.

${\displaystyle C_n}$ has anullspace of dimension 1, along the vector${\displaystyle J_{n,1}}$.

${\displaystyle C_n}$ is anorthogonal projection matrix. That is,${\displaystyle C_n\mathbf{v}}$ is a projection of${\displaystyle \mathbf{v}}$ onto the (n − 1)-dimensionalsubspace that is orthogonal to the nullspace${\displaystyle J_{n,1}}$. (This is the subspace of alln-vectors whose components sum to zero.)

The trace of${\displaystyle C_n}$ is${\displaystyle n(n-1)/n=n-1}$.

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 anm-by-n matrix${\displaystyle X}$.

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

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

${\displaystyle S=X{C}_n(X{C}_n{)}^{\mathrm{T}}=X{C}_n{C}_{n}X{}^{\mathrm{T}}=X{C}_{n}X{}^{\mathrm{T}}.}$

${\displaystyle C_n}$ is thecovariance matrix of themultinomial distribution, in the special case where the parameters of that distribution are${\displaystyle k=n}$, and${\displaystyle p_1=p_2=\cdots =p_n=\frac{1}{n}}$.

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