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

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

so that M is a (p+q)×(p+q) matrix.

Then Fischer's inequality states that

${\displaystyle \det(M)\leq \det(A)\det(C).}$

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 ${\displaystyle A^{-1}}$ and ${\displaystyle C^{-1}}$ are positive-definite. Let

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

We note that

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

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

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

By multiplicativity of determinant, we have

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

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

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

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

Taking the limit as ${\displaystyle \varepsilon \rightarrow 0}$ 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.

Improvements

If M can be partitioned in square blocks M_ij, then the following inequality by Thompson is valid:[1]

${\displaystyle \det(M)\leq \det([\det(M_{ij})])}$

where [det(M_ij)] is the matrix whose (i,j) entry is det(M_ij).

In particular, if the block matrices B and C are also square matrices, then the following inequality by Everett is valid:[2]

${\displaystyle \det(M)\leq \det {\begin{bmatrix}\det(A)&&\det(B)\\\det(B^{*})&&\det(D)\end{bmatrix}}}$

Thompson's inequality can also be generalized by an inequality in terms of the coefficients of the characteristic polynomial of the block matrices. Expressing the characteristic polynomial of the matrix A as

${\displaystyle p_{A}(t)=\sum _{k=0}^{n}t^{n-k}(-1)^{k}\operatorname {tr} (\Lambda ^{k}A)}$

and supposing that the blocks M_ij are m x m matrices, the following inequality by Lin and Zhang is valid:[3]

${\displaystyle \det(M)\leq \left({\frac {\det([\operatorname {tr} (\Lambda ^{r}M_{ij}]))}{\binom {m}{r}}}\right)^{\frac {m}{r}},\quad r=1,\ldots ,m}$

Note that if r = m, then this inequality is identical to Thompson's inequality.

See also

• Hadamard's inequality

Notes

1. Thompson, R. C. (1961). "A determinantal inequality for positive definite matrices". Canadian Mathematical Bulletin. 4: 57–62. doi:10.4153/cmb-1961-010-9.
2. Everitt, W. N. (1958). "A note on positive definite matrices". Glasgow Mathematical Journal. 3 (4): 173–175. doi:10.1017/S2040618500033670. ISSN 2051-2104.
3. Lin, Minghua; Zhang, Pingping (2017). "Unifying a result of Thompson and a result of Fiedler and Markham on block positive definite matrices". Linear Algebra and Its Applications. 533: 380–385. doi:10.1016/j.laa.2017.07.032.

References

• Fischer, Ernst (1907), "Über den Hadamardschen Determinentsatz", Arch. Math. U. Phys. (3), 13: 32–40.
• Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis, doi:10.1017/cbo9781139020411, ISBN 9781139020411.