Since the entries of the matrix are ( linear or constant ) polynomials in λ, the determinant is also a polynomial in λ.

Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL ( 2n, F ).

Since v is non-zero, this means that the matrix I − A is singular ( non-invertible ), which in turn means that its determinant is 0.

Since for any triangular matrix A the matrix, whose determinant is the characteristic polynomial of A, is also triangular, the diagonal entries of A in fact give the multiset of eigenvalues of A ( an eigenvalue with multiplicity m occurs exactly m times as diagonal entry ).

Since LaVeist's studies, segregation has received increased attention as a determinant of race disparities in mortality.

Since the adhesion of several cancer cells to endothelium requires the presence of endothelial selectins as well as sialyl Lewis carbohydrates on cancer cells, the degree of expression of selectins on the vascular wall and the presence of the appropriate ligand on cancer cells are determinant for their adhesion and extravasation into a specific organ.

Since U is unitary, it preserves the inner product:

Since 1996, when Avon was abolished, Bath has been the principal centre of the unitary authority of Bath and North East Somerset ( B & NES ).

Since the sum of the absolute squares of the amplitudes must be constant, must be unitary:

Since 1971 it is a municipality of unitary type, like all other municipalities of Sweden.

Since 1971 Haparanda, like all other municipalities of Sweden, is a municipality of unitary type.

Since U and V * are unitary, the columns of each of them form a set of orthonormal vectors, which can be regarded as basis vectors.

Since the center of U ( n ) is a 1-dimensional abelian normal subgroup of U ( n ), the unitary group is not semisimple.

Since T is a unitary operator, its spectrum lie on the unit circle.

Since the enactment of the Local Government and Rating Act 1997, district and unitary councils may create a parish council for a new civil parish either through a review or in response to a petition.

Since 13 February 2008 the power to create new parishes and parish councils, to alter parish boundaries, to dissolve parish councils and to abolish parishes has been devolved to district, unitary and London Borough councils ( collectively known as " principal councils ").

Since unitary representations are much easier to handle than the general case, it is natural to consider unitarizable representations, those that become unitary on the introduction of a suitable complex Hilbert space structure.

Since Cornwall became a unitary authority in 2009 there has been a two-tier structure of local government: Cornwall Council ( administers, for example, schools and highways, housing, social services, canal and harbour, refuse and recycling collection, street cleanliness ); and Bude-Stratton Town Council ( local children's playground, Bude " castle ").

Since Powys became a unitary authority in 1996, Radnorshire has been one of three areas formed under a decentralisation scheme.

Since the Lorentz group has no non-trivial unitary representation of finite dimension, it naively seems that one cannot construct a state with finite, non-zero spin and positive, Lorentz-invariant norm.

Since the first election to the unitary authority in 1995, the council has either been under Conservative party control, or no party has held a majority.

Since 1998 it has been part of the unitary authority of Windsor and Maidenhead.

Since that early time, however, other researchers ( e. g., Cattell, Guilford, and Thurstone ) have argued that intellect cannot be expressed in such a unitary manner, and have suggested more multifaceted approaches to intelligence.

Since April 2009 the county council has been a unitary authority.

Since any representation of a compact group is equivalent to a unitary representation, we take

Since the merger of the City Council into the Durham County Council unitary authority in 2009, mayoral appointments have been made by the Charter Trustees of the City of Durham ( composing the Durham County Councillors with divisions within the former City of Durham district area ).

Since the mid 1990s a series of local government reorganisations has seen a reduction in the number of county councils as unitary authorities have been established in a number of areas.

Since 1996 Wales has been divided into unitary principal areas.

Since the Lorentz group of special relativity is not compact, will not be unitary, so.

Since the entries of the Jacobian matrix are partial derivatives, we may simplify the above formula to get:

Since there are only four distinct eigenvalues for this matrix, they have some multiplicity.

Since the above is a linear system of equations ( more technically a linear transformation ), they can be written in matrix form:

Since the gels are fragile and the probes are unable to enter the matrix, the RNA samples, now separated by size, are transferred to a nylon membrane through a capillary or vacuum blotting system.

Since the column vectors of A are the row vectors of the transpose of A ( denoted here by A < sup > T </ sup >), column rank of A equals row rank of A is equivalent to saying that the rank of a matrix is equal to the rank of its transpose, i. e. rk ( A ) = rk ( A < sup > T </ sup >).

Since the 1990s, many physicists believe that 11-dimensional M-theory, which is described in many sectors by matrix string theory, in many other sectors by perturbative string theory, is the theory of everything.

Since the kth leading principal minor of a triangular matrix is the product of its diagonal elements up to row k, Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive.

Since matrix schemes follow the same laws of geometric progression as pyramids, they are subsequently as doomed to collapse.

Since the head is facing a different direction, the rotation matrix would be popped off the top of the stack and a different rotation matrix about the y-axis with a different angle would be pushed.

Since it was quite easy to stack interconnections ( wires ) inside the embedding matrix, the approach allowed designers to forget completely about the routing of wires ( usually a time-consuming operation of PCB design ): Anywhere the designer needs a connection, the machine will draw a wire in straight line from one location / pin to another.

Since is a rotation matrix its inverse is its transpose.

Since W ( by definition of the SVD of a real matrix ) is an orthogonal matrix, each row of Y < sup > T </ sup > is simply a rotation of the corresponding row of X < sup > T </ sup >.

Since < math > b_j < p </ math >, the elements of matrix are no other than the set of numbers:

Since an elementary reflection in the form of a Householder matrix can reduce any orthogonal matrix to this constrained form, a series of such reflections can bring any orthogonal matrix to the identity ; thus an orthogonal group is a reflection group.

Since the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together, the power k of a diagonal matrix A will have entries raised to the power.

Since shorter strands of DNA move through the gel matrix faster they are detected sooner and there is then a direct correlation between length of DNA strand and time at the detector.

Since the eigenvalues of this matrix coincide with the roots of the polynomial, one can use any eigenvalue algorithm to find the roots of the polynomial.

Since a shape memory polymer resin is used as the matrix, these composites have the ability to be easily manipulated into various configurations when they are heated above their activation temperatures and will exhibit high strength and stiffness at lower temperatures.

Since B is also a matrix with polynomials in t as entries, one can for each i collect the coefficients of in each entry to form a matrix B < sub > i </ sub > of numbers, such that one has