Determinant Sentence Examples
Consideration of the definition of the determinant shows that the value is unaltered when the suffixes in each element are transposed.
Indeed, while many diseases and health conditions are imprinted in our genetic code, our environment is a critical determinant of its unfolding.
If the determinant is transformed so as to read by columns as it formerly did by rows its value is unchanged.
No member of a determinant can involve more than one element from the first row.
The adjoint determinant is the (n - I) th power of the original determinant.Advertisement
Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.
Bezout's method gives the resultant in the form of a determinant of order m or n, according as m is n.
Such an expression as a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1' is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on.
Y ...a n v, the summation being for all permutations of the n numbers, is called the determinant of the n 2 quantities.
Each row as well as each column supplies one and only one element to each member of the determinant.Advertisement
Hence the transposition of columns merely changes the sign of the determinant.
Similarly it is shown that the transposition of any two columns or of any two rows merely changes the sign of the determinant.
Interchange of any two rows or of any two columns merely changes the sign of the determinant.
From the value of A we may separate those members which contain a particular element a ik as a factor, and write the portion aik A ik; A k, the cofactor of ar k, is called a minor of order n - i of the determinant.
This determinant and that associated with Aik are termed corresponding determinants.Advertisement
When a skew symmetric determinant is of even degree it is a perfect square.
Let the determinant of the b's be Ab and B rs, the minor corresponding to b rs .
We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.
We can prove that if the three equations be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant.
Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (yz),...Advertisement
If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x 2, -x 1 for a 1, a 2, and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on.
The second evectant is obtained by similarly operating upon all the symbols remaining which only occur in determinant factors, and so on for the higher evectants.
Certain convariants of the quintic involve the same determinant factors as appeared in the system of the quartic; these are f, H, i, T and j, and are of special importance.
But an attribute, though real, is not a distinct reality, but only a determinant of a substance, and has no being of its own apart from the substance so determined; whereas a substance, determined by all its attributes, is different from everything else in the world.
Colour, therefore, must be correlated with some determinant (determining factor) for pattern, and it cannot, therefore, exist alone in an animal's coat.Advertisement
And we must conceive that each kind of pattern - the self, the spotted, the striped, the hooded and all others - has its own special determinant.
When an albino mouse, rat, guinea-pig or rabbit is crossed with either a pure self or pure pied-coloured form, the offspring are similar to, though not always exactly like, the coloured parent; provided, of course, that the albino is pure and is not carrying some colour or pattern determinant which is dominant to that of the coloured parent used.
This conflict arises not only from naturalization having been granted without the corresponding expatriation having been permitted, but also from the fact that birth on the soil was the leading determinant of nationality by feudal law, and still is so by the laws of England and the United States (jus soli), while the nationality of the father is its leading determinant in those countries which have accepted Roman principles of jurisprudence (jus sanguinis).
The physiography of the state is the evident determinant of its climate, fauna and flora.
The idea, inasmuch as it is a law of universal mind, which in particular minds produces aggregates of sensations called things, is a "determinant" (iripas ixov), and as such is styled "quantity" and perhaps "number" but the ideal numbers are distinct from arithmetical numbers.
Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).
We might infer from this that the intellect, so judging, is itself the proper and complete determinant of the will, and that man, as a rational being, ought to aim at the realization of absolute good for its own sake.
With Price, again, he holds that rightness of intention and motive is not only an indispensable condition or element of the rightness of an action, but actually the sole determinant of its moral worth; but with more philosophical consistency he draws the inference - of which the English moralist does not seem to have dreamt - that there can be no separate rational principles for determining the " material " rightness of conduct, as distinct from its " formal " rightness; and therefore that all rules of duty, so far as universally binding, must admit of being exhibited as applications of the one general principle that duty ought to be done for duty's sake.
Considering the equations ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.
It is to be noticed that the resulting equation is a,b,c x= d,b,c,, ,, a' b' c' d'b' c' an, b", cn d", b", c" where the expression on the right-hand side is the like function with d, d', d" in place of a, a', a" respectively, and is of course also a determinant.
The products in question may be obtained by permuting in every possible manner the columns (or the lines) of the determinant, and then taking for the factors the n elements in the dexter diagonal.
Thus, for three columns, it appears by either rule that 123, 231, 312 are positive; 213, 321, 132 are negative; and the developed expression of the foregoing determinant of the third order is =ab'c" - ab "c'+a'b "c - a'bc" - a"bc' - a"b'c. 3.
It further appears that a determinant is a linear function' of the elements of each column thereof, and also a linear function of the elements of each line thereof; moreover, that the determinant retains the same value, only its sign being altered, when any two columns are interchanged, or when any two lines are interchanged; more generally, when the columns are permuted in any manner, or when the lines are permuted in any manner, the determinant retains its original value, with the sign + or - according as the new arrangement (considered as derived from the primitive arrangement) is positive or negative according to the foregoing rule of signs.
It at once follows that, if two columns are identical, or if two lines are identical, the value of the determinant is = o.
It may be added, that if the lines are converted into columns, and the columns into lines, in such a way as to leave the dexter diagonal unaltered, the value of the determinant is unaltered; the determinant is in this case said to be transposed.
By what precedes it appears that there exists a function of the n 2 elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient + 1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations.
Observe that the properties show at once that if any column is = o (that is, if the elements in the column are each = o), then the determinant is = o; and further, that if any two columns are identical, then the determinant is = o.
Reverting to the system of linear equations written down at the beginning of this article, consider the determinant ax+by+cz - d,b,c a' x+b' y+c'z - d', b', c" a"x+b"y+c"z - d", b", c" it appears that this is viz.
It is most simply expressed thus where the expression on the left side stands for a determinant, the terms"of the first line being (a, b, c) (a, a', a"), that is, as+ ba'+ ca", (a, b, c) (/3, /3', 13"), that is, a/3+b/3'+0", (a, b, c) (y, y, 'Y'), that is ay+by'+cy"; and similarly the terms in the second and third lines are the life functions with (a', b', c') and (a", b",c") respectively.
To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be I The reason is the connexion with the corresponding theorem for the multiplication of two matrices.
Observe that for a determinant of the n-th order, taking the decomposition to be r + (n - I), we fall back upon the equations given at the commencement, in order to show the genesis of a determinant.
Any determinant I a,' b, I formed out of the elements of the original determinant, by selecting the lines and columns at pleasure, is termed a minor of the original determinant; and when the number of lines and columns, or order of the determinant, is n - I, then such determinant is called a first minor; the number of the first minors is = n 2, the first minors, in fact, corresponding to the several elements of the determinant - that is, the coefficient therein of any term whatever is the corresponding first minor.
The first minors, each divided by the determinant itself, form a system of elements inverse to the elements of the determinant.
Laplace developed a theorem of Vandermonde for the expansion of a determinant, and in 1773 Joseph Louis Lagrange, in his memoir on Pyramids, used determinants of the third order, and proved that the square of a determinant was also a determinant.
To Gauss is due the establishment of the important theorem, that the product of two determinants both of the second and third orders is a determinant.
The definition of a determinant in all dimensions will be given in detail, together with applications and techniques for calculating determinants.
They therefore represent a finite scientific and economic resource and are a notable determinant of landscape character.
The determinant of a square diagonal matrix is the product of its diagonal matrix is the product of its diagonal elements.
The fermion determinant is calculated by summing over the resulting complex eigenvalues.
The determinant of a permutation matrix equals the signature of the column permutation matrix equals the signature of the column permutation.
The determinant of a permutation matrix equals the signature of the column permutation.
A matrix has in many parts of mathematics a signification apart from its evaluation as a determinant.
If we multiply the elements of the second row by an arbitrary magnitude X, and add to the corresponding elements of the first row, A becomes Zai,A18+XEa28A13 = Lia13A18 =A, showing that the value of the determinant is unchanged.
If the jth column be identical with the i ll ' the determinant A vanishes identically; hence if j be not equal to i, k, or r, a 11 a 21 a31 0 =I alk a2k a3k A11.
Such a determinant when of uneven degree vanishes, for if we multiply each row by - I we multiply the determinant by (- I) n = -1, and the effect of this is otherwise merely to transpose the determinant so that it reads by rows as it formerly did by columns, an operation which we know leaves the determinant unaltered.
Making the substitution in any symbolic product the only determinant factors that present themselves in the numerator are of the form (af), (bf), (cf),...and every symbol a finally appears in the form.
He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n 2 differential coefficients of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations (see Algebraic Forms).
They carry only some determinant or determinants which are capable of developing colour when they interact with some other determinant or determinants carried alone by pigmented individuals.
The determinant is usually written all a12 a13.
If any two rows or any two columns of a determinant be identical the value of the determinant is zero.
Hence anAu = auk t a22a33...ann, where the cofactor of an is clearly the determinant obtained by erasing the first row and the first column.
Every factor common to all the elements of a row or of a column is obviously a factor of the determinant, and may be taken outside the determinant brackets.
The minor Aik is aa, and is itself a determinant of order n-t.
In particular the square of a determinant is a deter minant of the same order (b 11 b 22 b 33 ...b nn) such that bik = b ki; it is for this reason termed symmetrical.
Hence the product determinant has the principal diagonal elements each equal to A and the remaining elements zero.
The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.
It was observed above that the square of a determinant when expressed as a determinant of the same order is such that its elements have the property expressed by aik = aki.
It is easy to see that the adjoint determinant is also 'symmetrical, viz.
A skew symmetric determinant has a,.
In the case of the determinant of order 4 the square root is Al2A34 - A 13 A 24 +A14A23.
A skew determinant is one which is skew symmetric in all respects,.
Such a determinant is of importance in the theory of orthogonal substitution.
For the second order we may take Ob - I - A, 1 1 +A2, and the adjoint determinant is the same; hence (1 +A2)x1 = (1-A 2)X 1 + 2AX2, (l +A 2)x 2 = - 2AX1 +(1 - A2)X2.
By solving the equations of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1, x 2 to be independents, so that r must differ from zero.
The identities are, in particular, of service in reducing symbolic products to standard forms. A symbolical expression may be always so transformed that the power of any determinant factor (ab) is even.
Moreover, representing the remaining three lines by a" b" c" d" e" b /r c a, d N, e"' a " c 'N d"N err" it is further seen that the factor which multiplies the determinant formed with any two columns of the first set is the determinant of the third order formed with the complementary three columns of the second set; and it thus appears that the determinant of the fifth order is a sum of all the products of the form ' a b c" d" e" a, b"c"'dN, ear the sign being in each case such that the sign of the term .c"d"'e" obtained from the diagonal elements of the component determinants may be the actual sign of this term in the determinant of the fifth order; for the product written down the sign is obviously +.
The subject matter of the text is, obviously, a crucial determinant of the role that spatial inferences play in understanding it.
Coupled with the 24-hour pH probe study, the test becomes the best determinant of GERD because it actually monitors how often the patient has reflux into the esophagus during a full day.
Youth gangs are bound by a common ethnicity, race, social class, or other determinant and employ distinctive symbols, including style and color of dress, hand signs, tattoos, and graffiti.
Motivation to change risk behaviors is another determinant of prevention and affects whether a person acts on his or her knowledge of the transmission and prevention of HIV.
Behavioral skills for engaging in specific prevention behaviors are a third determinant of prevention; it affects whether a knowledgeable, highly motivated person will be able to change his or her behavior to prevent HIV.
Bleeding from the nose is the obvious determinant of a nosebleed.
A good determinant in making your decision is to remind yourself that if it sounds too good to be true, it probably is.
This helps preserve muscle mass and keep your natural metabolism going, which is a key determinant to your progress.
We thus obtain for the product a determinant of order n.
While cost is always an important determinant when shopping for insurance, it is also necessary to keep in mind just what type of financial risk you're attempting to shield yourself from.
This rate is used as a determinant for a company's possible growth.
In the original Star Trek, the hierarchy of officers was a key determinant of the big three.