Fischer's inequality

In mathematics, Fischer's inequality gives an upper bound for the determinant of a positive-semidefinite matrix whose entries are complex numbers in terms of the determinants of its principal diagonal blocks. Suppose A, C are respectively p×p, q×q positive-semidefinite complex matrices and B is a p×q complex matrix. Let

M:=\left[{\begin{matrix}A&B\\B^{*}&C\end{matrix}}\right]

so that M is a (p+q)×(p+q) matrix.

Then Fischer's inequality states that

\det(M)\leq \det(A)\det(C).

If M is positive-definite, equality is achieved in Fischer's inequality if and only if all the entries of B are 0. Inductively one may conclude that a similar inequality holds for a block decomposition of M with multiple principal diagonal blocks. Considering 1×1 blocks, a corollary is Hadamard's inequality.

Proof

Assume that A and C are positive-definite. We have $A^{-1}$ and $C^{-1}$ are positive-definite. Let

D:=\left[{\begin{matrix}A&0\\0&C\end{matrix}}\right].

We note that

D^{-{\frac {1}{2}}}MD^{-{\frac {1}{2}}}=\left[{\begin{matrix}A^{-{\frac {1}{2}}}&0\\0&C^{-{\frac {1}{2}}}\end{matrix}}\right]\left[{\begin{matrix}A&B\\B^{*}&C\end{matrix}}\right]\left[{\begin{matrix}A^{-{\frac {1}{2}}}&0\\0&C^{-{\frac {1}{2}}}\end{matrix}}\right]=\left[{\begin{matrix}I_{p}&A^{\frac {1}{2}}BC^{-{\frac {1}{2}}}\\C^{-{\frac {1}{2}}}B^{*}A^{-{\frac {1}{2}}}&I_{q}\end{matrix}}\right]

Applying the AM-GM inequality to the eigenvalues of $D^{-{\frac {1}{2}}}MD^{-{\frac {1}{2}}}$ , we see

\det(D^{-{\frac {1}{2}}}MD^{-{\frac {1}{2}}})\leq \left({1 \over p+q}\mathrm {tr} (D^{-{\frac {1}{2}}}MD^{-{\frac {1}{2}}})\right)^{p+q}=1^{p+q}=1.

By multiplicativity of determinant, we have

{\begin{aligned}\det(D^{-{\frac {1}{2}}})\det(M)\det(D^{-{\frac {1}{2}}})\leq 1\\\Longrightarrow \det(M)\leq \det(D)=\det(A)\det(C).\end{aligned}}

In this case, equality holds if and only if M = D that is, all entries of B are 0.

For $\varepsilon >0$ , as $A+\varepsilon I_{p}$ and $C+\varepsilon I_{q}$ are positive-definite, we have

\det(M+\varepsilon I_{p+q})\leq \det(A+\varepsilon I_{p})\det(C+\varepsilon I_{q}).

Taking the limit as $\varepsilon \rightarrow 0$ proves the inequality. From the inequality we note that if M is invertible, then both A and C are invertible and we get the desired equality condition.

References

Fischer, Ernst (1907), "Über den Hadamardschen Determinentsatz", Arch. Math. u. Phys. (3), 13: 32–40 .

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Fischer's inequality

Proof

See also

References