Determinant problem for block-matrices Torsten Sillke, 1995-12-28
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Let A, B, C, D be n*n square matrices.
| A B |
| C D | = | A*D - C*B |
if A*C = C*A (**).
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Typically the proof for this result is given as follows:
Case I) A is invertible.
Let A' = Inverse(A).
( I 0 ) ( A B ) ( A B )
( -C*A' I ) * ( C D ) = ( 0 -C*A'*B+D )
| A B | | A B |
| C D | = | 0 -C*A'*B+D | = |A| * | -C*A'*B+D |
= | A*D - A*C*A'B | = | A*D - C*B |
Case II) otherwise
Perturbation method:
Replace A by A + epsilon*I.
Now (**) is valid for the perturbated A too.
So we are in case I again. Now take the limit epsilon -> 0.
Problem:
Can one avoid this analytic method?
Is there a pure algebraic derivation?
In other words, does the equation hold for all commutative rings?
Solution:
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Torsten Sillke, William P. Wardlaw;
Q873 Quickie on Block Determinants,
Mathematical Magazin, 70:5 (Dec. 1997) 382
References:
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- I. Kovacs, D. S. Silver, S. G. Williams,
Determinants of Commuting-Block Matrices,
American Mathematical Monthly 106 (Dec. 1999) 950-952
- M. Marcus,
Two determinant condensation formulas
Linear and Multilinear Algebra 22 (1987), no. 1, 95--102.
- L. Mirsky,
Linear Algebra, 1955, Clarendon Pr. Oxford
Chapter III Problem 36, p110
- AMM 82 (Nov. 1975) 942, Adv. Problem 6057, Determinant of Matrices
- AMM 84 (June-July 1977) 495-496, Adv. Prob. 6057 Sol. Determinant of Matrices
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89g:15009 15A15
Marcus, Marvin(1-UCSB-C)
Two determinant condensation formulas. (English)
Linear and Multilinear Algebra 22 (1987), no. 1, 95--102.
Let $A,B,C,D$ be square matrices of order $n$ with entries from an
integral domain $R$ with 1 and of characteristic not 2. If $CD\sp
T=DC\sp T$ and $D$ is nonsingular then $\det \left[\smallmatrix A&B\
C&D\endsmallmatrix\right]= \det(AD\sp T-BC\sp T)$. Similarly if $CD\sp
T=-DC\sp T$ and $D$ is nonsingular then $\det\left[\smallmatrix A&B\
C&D\endsmallmatrix \right]=\det(AD\sp T+BC\sp T)$. The first result
remains true when $D$ is singular but the second one fails. The
case in which $R$ is the real or complex field and $D$ is singular
have been treated with continuity arguments. The author proves by
matrix methods for arbitrary $R$ that the first equality holds and
the second holds if and only if $n-\roman{rank}(D)$ is even.
Reviewed by M. Pearl
87c:15001 15-01 15A24
Wardlaw, William P.(1-USNA)
A transfer device for matrix theorems. (English)
Math. Mag. 59 (1986), no. 1, 30--33.
Let $F$ be a formula---such as the Cayley-Hamilton theorem or the product
formula for determinants---involving matrices over a commutative ring.
The author shows how to establish several such formulas $F$ over an
arbitrary commutative ring once one knows them in the special case of
matrices of real numbers, perhaps invertible or with distinct eigenvalues.
The proofs make use of the fact that the real numbers have
infinite transcendence degree over the rational numbers.
The motivation is to make graduate algebra classes more interesting by
enabling continuity to be used in the proof of theorems about arbitrary
commutative rings. (For another interesting idea of this type, see a paper
by W. Watkins [Amer. Math. Monthly 82 (1975), 364--368; MR 51 #554].)
Reviewed by Lawrence S. Levy