M1, M2, M3, and M4 are four tri-diagonal non-singular sub-matrices and X is the big matrix that contain M1, M2, M3, and M4.
please, i need reference to show the matrix X is also non-singular.
The big matrix X is not necessary non-singular, because we can find a big matrix with four sub-matrices are non-singular but the matrix X has two parallel rows are identical entries which det(X) = 0.
Thank you very much, in my work M1 not equal M3 and M2 not equal M4. All the matrices are different, square , tri-diagonal, and non-singular. and also i am sure this matrix is non-singular, but i need to know how to prove that ?
If there are some theorem or definition, please.
Proposition (Schur complement formula). Let M ∈ M_{n}(F) read blockwise
M=[
A B
C D
]
where the diagonal blocks are square and A is invertible. Then
detM = detA.det(D - CA⁻¹B).
The same (as Wiwat): please write "Schur complement" in Google,
you will find many useful links! Regards, Lajos.
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