Word Finder - determinant

definition

A determining factor; an element that determines the nature of something.

definition

A scalar that encodes certain characteristics of a given transformation matrix; the unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value 1 for the unit matrix; abbreviated as: det.

definition

A substance that causes a cell to adopt a particular fate.

definition

Serving to determine or limit; determinative.

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.

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.

The adjoint determinant is the (n - I) th power of the original determinant.

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.

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.

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.

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 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),...

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.

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.