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
so that M is a (p+q)×(p+q) matrix.
Then Fischer's inequality states that
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. On the other hand, Fischer's inequality can also be proved by using Hadamard's inequality, see the proof of Theorem 7.8.5 in Horn and Johnson's Matrix Analysis.
Proof
Assume that A and C are positive-definite. We have and are positive-definite. Let
We note that
Applying the AM-GM inequality to the eigenvalues of , we see
By multiplicativity of determinant, we have
In this case, equality holds if and only if M = D that is, all entries of B are 0.
For , as and are positive-definite, we have
Taking the limit as 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]
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]
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
and supposing that the blocks Mij are m x m matrices, the following inequality by Lin and Zhang is valid:[3]
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.