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 \( A^{-1} \) 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 \( \varepsilon >0 \), 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\( \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 Mij, then the following inequality by Thompson is valid:[1]
\( {\displaystyle \det(M)\leq \det([\det(M_{ij})])} \)
where [det(Mij)] is the matrix whose (i,j) entry is det(Mij).
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 Mij 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
Thompson, R. C. (1961). "A determinantal inequality for positive definite matrices". Canadian Mathematical Bulletin. 4: 57–62. doi:10.4153/cmb-1961-010-9.
Everitt, W. N. (1958). "A note on positive definite matrices". Glasgow Mathematical Journal. 3 (4): 173–175. doi:10.1017/S2040618500033670. ISSN 2051-2104.
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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