If the LU decomposition exists, the LDU decomposition does too.

If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P = I, so an LU decomposition exists.

If the direct-sum decomposition ( A ) is valid, how can we get hold of the projections Af associated with the decomposition??

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

Statements such as the Banach – Tarski paradox can be rephrased as conditional statements, for example, " If AC holds, the decomposition in the Banach – Tarski paradox exists.

If sugars are heated so that all water of crystallisation is driven off, then caramelization starts, with the sugar undergoing thermal decomposition with the formation of carbon, and other breakdown products producing caramel.

If a decomposition exists with each c < sub > i </ sub > a centrally primitive idempotent, then R is a direct sum of the corner rings c < sub > i </ sub > Rc < sub > i </ sub >, each of which is ring irreducible.

If the fire was small, however, the convict would burn for some time until death from heatstroke, shock, the loss of blood and / or simply the thermal decomposition of vital body parts.

If I is a homogeneous ideal in A, then is also a graded ring, and has decomposition

If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an eigenvalue decomposition determined by solving the system of equations

If the n eigenvalues are distinct ( that is, none is equal to any of the others ), then V is invertible, implying the decomposition.

If is the singular value decomposition of, then.

If the protocol is unconditionally concealing, then Alice can unitarily transform these states into each other using the properties of the Schmidt decomposition, effectively defeating the bindingness property.

If the matrix X < sup > T </ sup > X is well-conditioned and positive definite, that is, it has full rank, the normal equations can be solved directly by using the Cholesky decomposition R < sup > T </ sup > R, where R is an upper triangular matrix, giving:

If the measure ' is complex-valued i. e. is a complex measure, its upper and lower variation cannot be defined and the Hahn – Jordan decomposition theorem can only be applied to its real and imaginary parts.

If i ≠ j, then this decomposition does not contain the trivial representation ( Otherwise, we'd have a nonzero intertwiner from V < sub > j </ sub > to V < sub > i </ sub > contradicting Schur's lemma ).

If a split-complex number z does not lie on one of the diagonals, then z has a polar decomposition.

If they settle to the bottom, phytoplankton release ammonia during their decomposition, which returns nitrogen to the waters.

If the vacuum is degenerate and we have a mixed state, the cluster decomposition property fails.

If an integer n has prime factorization, then the primary decomposition of the ideal generated by, is

If cyanogen iodide is heated enough to undergo complete decomposition, it may releases toxic fumes of nitrogen oxides, cyanide and iodide.

If a matrix A is diagonalizable, the space can be decomposed into a direct sum of eigenspaces of A, and the matrix has an eigenspace decomposition.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If he found data that fitted his general plan, he used it and counted his sources trustworthy.

If they are used, the walls and roof should be 10 inches thick to give the same protection as the 8-inch solid concrete blocks.

If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.

Note: If 1/2-inch panel board is used inside and out, or 5/8-inch one side and 3/8-inch the other, and 1/8-inch glass is used, stock lumber in Af, Af, and Af can be used in making the glass panels.

If only state funds were used to pay for the vocational education, it could be argued that the state should not have to bear the cost of vocational training which would benefit employers in other states.

If the master of scops who was most responsible for the poem ever used kennings that were traditional, he was at least partly deprived of free will and not inclined towards shrewd and sophisticated misuse of speech elements.

If the one- or two-digit address of an index word or electronic switch is used or is included in the operand of an XRESERVE or SRESERVE statement ( see page 99 ), the corresponding index word or electronic switch is reserved.

If the voice is just a shade less glorious than it used to be, it is still a beautiful instrument, controlled and flexible.

If, for instance, someone says " I buried $ 100 in the bank ", most people would not think someone used a shovel to dig in the mud.

If " biweekly " is used in a conversation about a meeting schedule, it may be difficult to infer which meaning was intended.

If the Greek letter is used, it is assumed to be a Fourier transform of another function,

If the latter, Amaryllis would be the correct name for the genus Hippeastrum, and a different name would have to be used for the genus discussed here.

If an alpha channel is used in an image, it is common to also multiply the color by the alpha value, to save on additional multiplications during compositing.

If this article / noun pair is used as the object of a verb, it ( usually ) changes to the accusative case, which entails an article shift in German – Ich sehe den Wagen.

If, conversely, tin is used to coat steel, when a breach of the coating occurs it actually accelerates oxidation of the iron.

If the amount in the sample is below an instrument's range of measurement, the method of addition can be used.

If an ESN / MIN Pair is intercepted, it could then be cloned onto a different phone and used in other areas for making calls without paying.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If used for drinking, mineral supplements or mineralization is necessary.

If the assets used are not identical ( so a price divergence makes the trade temporarily lose money ), or the margin treatment is not identical, and the trader is accordingly required to post margin ( faces a margin call ), the trader may run out of capital ( if they run out of cash and cannot borrow more ) and go bankrupt even though the trades may be expected to ultimately make money.

If chosen, the SysOp could then initiate a text-to-text chat with the caller ; similar to what commercial websites have used to sell and support products.

If either one die or the other but not both can be moved, the higher must be used.

If F. tularensis were used as a weapon, the bacteria would likely be made airborne for exposure by inhalation.