Hawking energy

In linear algebra, a constrained generalized inverse is obtained by solving a system of linear equations with an additional constraint that the solution is in a given subspace. One also says that the problem is described by a system of constrained linear equations.

In many practical problems, the solution $x$ of a linear system of equations

$Ax=b(\text{with given }A\in {\mathbb{ℝ}}^{m\times n}\text{ and }b\in {\mathbb{ℝ}}^m)$

is acceptable only when it is in a certain linear subspace $L$ of ${\mathbb{ℝ}}^m$.

In the following, the orthogonal projection on $L$ will be denoted by $P_L$. Constrained system of linear equations

$Ax=bx\in L$

has a solution if and only if the unconstrained system of equations

$(A{P}_L)x=bx\in {\mathbb{ℝ}}^m$

is solvable. If the subspace $L$ is a proper subspace of ${\mathbb{ℝ}}^m$, then the matrix of the unconstrained problem $(A{P}_L)$ may be singular even if the system matrix $A$ of the constrained problem is invertible (in that case, $m=n$). This means that one needs to use a generalized inverse for the solution of the constrained problem. So, a generalized inverse of $(A{P}_L)$ is also called a $L$-constrained pseudoinverse of $A$.

An example of a pseudoinverse that can be used for the solution of a constrained problem is the Bott-Duffin inverse of $A$ constrained to $L$, which is defined by the equation

$A_L^{(-1)}:=P_L(A{P}_L+P_{L^{\perp }}{)}^{-1},$

if the inverse on the right-hand-side exists.

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