Approximation of a matrix multiplication

Question

Let $A_{n \times n}$ be a positive-definite matrix. There is a way to approximate the following product?

$$\left(I-\frac{1}{n}11^{T}\right) A \left(I-\frac{1}{n}11^{T}\right)$$

where $1$ is a column vector of ones.

Answer 1

Let $U_1:=\dfrac{1}{\sqrt{n}}\boldsymbol{1}$ the normalized version of $\boldsymbol{1}$.

Let $\Pi=\mathbb{R}U_1^{\perp}$ (hyperplane orthogonal to vector $V$).

Let $\{U_2,U_3,\cdots U_n\}$ be vectors complementing $U_1$ in such a way that they constitute an ortho-normal basis of $\mathbb{R^n}$ (each vector is orthogonal to all others ; all with unit norms).

Now comes the geometric interpretation :

• $U_1U_1^T$ is tha matrix of the orthogonal projection onto one-dimensional subspace $\mathbb{R}U_1$.

• Therefore, $$P:=I-\frac1n \boldsymbol{11}^T=I-U_1U_1^T$$ is the matrix of the orthogonal projection onto hyperplane $\Pi$.

As a consequence, $P^T=P$ ;

The computation of the value of the quadratic form

$$X^T(PAP)X=(PX)^TA(PX)$$

is based on the efficiency of computation of $PX$.

This can be done by expressing the classical decomposition of a vector onto a subspace and its complementary subspace :

$$X=U_1(U_1^T.X)+V$$ where $V \in \Pi$, giving

$$PX=\underbrace{PU_1}_{= \ 0}(U_1^T.X)+\underbrace{PV}_{= \ V}.$$