Sentence examples for matrix it follows from inspiring English sources

Note that, for any diagonal matrix it follows that (37).

where is a unitary matrix, it follows that the solution of the system (2.8) can be written as (2.10).

For cases (H3)(1) and (H3)(4), since A is a positive semidefinite matrix, it follows from (1.5) that u k is nonconstant.

end{aligned} (6) Since (M_{1}) is a P-matrix and Ω is a positive diagonal matrix, it follows from Lemma 3 that (M_{1}+Omega) is a nonsingular matrix.

(6) Since (lambdaleq0) and (D_{1}^{(x)}) is a positive diagonal matrix, it follows from Lemma 1 that (A+D_{1}^{(x)}-lambda I) is an irreducible nonsingular M-matrix.

Next, consider the dynamical system dot{x}(t)=-L_{2}^{mathrm{I}}x t)+tilde{f}^{mathrm{I}}_{2max},quad x 0)=tilde {S}_{20}^{mathrm{I}},quad tgeq0, (51) and note that since (L_{2}^{mathrm{I}}) is a diagonal positive-definite matrix, it follows that (x t)) is bounded for all (tgeq0).

Multiplying both sides of P y ̃ y ̃ by the matrices U ̃ p H and U ̃ s and using the orthogonality property between these matrices, it follows, U ̃ p H P y ̃ y ̃ U ̃ s = 0 (3.2). that we can write as, ( U p + Δ U p ) H ( P y ° y ° + Δ P yy ) ( U s + Δ U s ) = 0 (3.3).

Since (S_{1}) and (S_{2}) are real positive diagonal matrices, it follows from Assumption (A2) that begin{aligned}& 0 leqtilde{q}^(t)Gamma S_{1}Gammatilde{q}(t) -f^ bigl tilde{q}(t bigr S_{1} fbigl tilde{q}(t bigr), end{aligned} (23) begin{aligned}& 0 leqtilde{q}^ t-tau Gamma S_{2}Gammatilde{q}(t-tau)-f^ t-tau Gamma{q}(t-tau)bigr)S_{2}Gammatilde{q} t-tau -f^bigl.

Since both Q2(l ) and Q2(r ) are tridiagonal matrices, it follows from (13) that Q2(12) is a block tridiagonal matrix.

As it is the cross-linked structure that imparts strength and integrity to intact basement membrane matrices, it follows that Matrigel matrices would have lower resistance to cell penetration and be unable to reflect the in vivo situation.

where λ min ( P ) is the minimum eigenvalue of the matrix P. It follows from (26) that ∥ x ( t ) ∥ < x T ( 0 ) P x ( 0 ) λ min ( P ) exp ( κ t ) = x T ( 0 ) P x ( 0 ) λ min ( P ) exp ( − κ 2 t ).