Consider

$$C = A^H D A + M$$

where

• $A$ is a $m \times m$ unitary matrix.

• $D$ is a $m \times m$ diagonal matrix with entries either $0$ or $1$. The number of $1$'s is $n \ll m$.

• $M$ is a $m \times m$ diagonal matrix with all real non-negative entries.

It is known that $C$ is a symmetric positive definite matrix. Is there a fast way to compute the lowest eigenvalue (need not compute the eigenvector) of $C$?

Especially given $n \ll m$ and $m$ being very large I cannot afford to compute all $m$ eigenvalues. Also I would like to avoid storing a $m \times m$ matrix in memory if possible.

PS : Inverse iteration, or Power iteration methods are generic and I don't find them taking any advantage of this specific scenario. So is Rayleigh Quotient method. Appreciate some leads for an efficient algorithm.

Consider

$$C = A^H D A + M$$

where

• $A$ is a $m \times m$ unitary matrix.

• $D$ is a $m \times m$ diagonal matrix with entries either $0$ or $1$. The number of $1$'s is $n \ll m$.

• $M$ is a $m \times m$ diagonal matrix with all real non-negative entries, with atleast one positive entry.

It is known that $C$ is a symmetric positive definite matrix. Is there a fast way to compute the lowest eigenvalue (need not compute the eigenvector) of $C$?

Especially given $n \ll m$ and $m$ being very large I cannot afford to compute all $m$ eigenvalues. Also I would like to avoid storing a $m \times m$ matrix in memory if possible.

PS : Inverse iteration, or Power iteration methods are generic and I don't find them taking any advantage of this specific scenario. So is Rayleigh Quotient method. Appreciate some leads for an efficient algorithm.

Consider

$$C = A^H D A + M$$

where

• $A$ is a $m \times m$ unitary matrix.

• $D$ is a $m \times m$ diagonal matrix with entries either $0$ or $1$. The number of $1$'s is $n \ll m$.

• $M$ is a $m \times m$ diagonal matrix with all non-negative entries.

It is known that $C$ is a symmetric positive definite matrix. Is there a fast way to compute the lowest eigenvalue (need not compute the eigenvector) of $C$?

Especially given $n \ll m$ and $m$ being very large I cannot afford to compute all $m$ eigenvalues. Also I would like to avoid storing a $m \times m$ matrix in memory if possible.

PS : Inverse iteration, or Power iteration methods are generic and I don't find them taking any advantage of this specific scenario. So is Rayleigh Quotient method. Appreciate some leads for an efficient algorithm.

Consider

$$C = A^H D A + M$$

where

• $A$ is a $m \times m$ unitary matrix.

• $D$ is a $m \times m$ diagonal matrix with entries either $0$ or $1$. The number of $1$'s is $n \ll m$.

• $M$ is a $m \times m$ diagonal matrix with all real non-negative entries.

It is known that $C$ is a symmetric positive definite matrix. Is there a fast way to compute the lowest eigenvalue (need not compute the eigenvector) of $C$?

Especially given $n \ll m$ and $m$ being very large I cannot afford to compute all $m$ eigenvalues. Also I would like to avoid storing a $m \times m$ matrix in memory if possible.

PS : Inverse iteration, or Power iteration methods are generic and I don't find them taking any advantage of this specific scenario. So is Rayleigh Quotient method. Appreciate some leads for an efficient algorithm.