Matrix Inverse %. 2 _ 2 Matrix Divide If y is a non-singular matrix, then %.y is the inverse of y. For example: mp=. +/ . * Matrix product (%. ; ] ; (%. mp ]))i.2 2õııııııııÃıııÃıııÀş_1.5 0.5ş0 1ş1 0şş 1 0ş2 3ş0 1şÁııııııııÂıııÂıııãMore generally, %.y is defined in terms of the dyadic case, with the left argument =i.{:$y (an identity matrix) or, equally, by the relation(%.y)mp xx %. y . The shape of %.y is |.$y. The degenerate vector and scalar cases are defined by using the matrix ,.y, but the shape of the result is $y. F or a non-zero vector y, the result of %.y is a vector collinear with y whose length is the reciprocal of that of y; it is called the reflection of y in the unit circle (or sphere). Thus: (%.,:(] % %.)) y=.2 3 4 0.0689655 0.103448 0.137931 29 29 29 If y is non-singular, then x %. y is (%.y) mp x. More generally, if the columns of y are linearly independent and if #x and #y agree, then x %. y minimizes the difference: d=. x - y mp x %. y in the sense that the magnitudes +/ d*+d are minimized. Degenerate cases of y are treated as the one-column matrix ,.y.Geometrically, y mp x %. y is the projection of the vector x on the column space of y; the point nearest to x in the space spanned by the columns of y. Common uses of %. are in the solution of linear equations, and in the approximation of functions by polynomials, as in the expression c=.á(f x)%. x ^ / i.4.We will illustrate the use of %. in function fitting by the sine function, showing, in particular, the maxim um over the magnitudes of the differences from the function being approximated: sin =. 1&o. [. x=. 5 %~ i. 6 Function to be approximated c=. (sin x) %. x ^/ i. 4 Use of matrix divide ,.&.>@(] ; c"_ ; sin ; c&p. ; >./@|@(sin-c&p.)) x õıııÃıııııııııııÃıııııııııÃıııııııııııÃııııııııııııÀ ş 0ş_5.30503e_5ş 0ş_5.30503e_5ş0.0001679919ş ş0.2ş 1.00384ş0.1986693ş 0.1988263ş ş ş0.4ş_0.01845296ş0.3894183ş 0.3893211ş ş ş0.6ş _0.1439224ş0.5646425ş 0.5645231ş ş ş0.8ş ş0.7173561ş 0.7175241ş ş ş 1ş ş 0.841471ş 0.8414157ş ş ÁıııÂıııııııııııÂıııııııııÂıııııııııııÂııııııııııııã Square Root %: _ 0 0 Root %: y is the square root of y. If y is negative, the result is an imaginary number. For example, %:-4 is 0j2. x %: y is the x root of y. Thus, 3á%: 8 is 2, and 2%:y is %:y. In general,x %: y is y^%x. For example: y=. i. 7 y 0 1 2 3 4 5 6 2 %: y 0 1 1.41421 1.73205 2 2.23607 2.44949 %: y 0 1 1.41421 1.73205 2 2.23607 2.44949 r=. 1 2 3 4 z=. r %:/ y z 0 1 2 3 4 5 6 0 1 1.41421 1.73205 2 2.23607 2.44949 0 1 1.25992 1.44225 1.5874 1.70998 1.81712 0 1 1.18921 1.31607 1.41421 1.49535 1.56508 r ^~ z See agreement in Section II B, and note use of ~ . 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Exponential ^ _ 0 0 Power ^y is equivalent to e^y, where e is Euler's number ^1 (approximately 2.71828). The natural logarithm (^.) is inverse to ^ (that is, y=^.^y and y=^^.y).The monad x&^ is inverse to the monad x&^. . For example: 10&^. ^:_1õııÃıÃıÀş10ş&ş^şÁııÂıÂıã 10&^ 10&^. 1 2 3 4 51 2 3 4 5 10&^. 10&^ 1 2 3 4 51 2 3 4 5 x^2 and x^3 and x^0.5 are the square, cube, and square root of x.The general definition of x^y is ^y*^.x, applying for complex numbers as well as real. For a non-negative integer y, the ph rase x ^ y is equivalent to */y # x; in particular, */ on an empty list is 1, and x^0 is 1 for any x, including 0.The fit conjunction applies to ^ to yield a stope defined as follows: xá^!.k n is */x + k*i. n. In particular, ^!._1 is the falling factorial function. The last result in the first example below illustrates the falling factorial function, formed by the fit conjunction. See Chapter 4 of [9] for the use of stope functions, stope polynomials, and Stirling numbers in the difference calculus: e=. ^ 1 [ x=. 4 [ y=. 0 1 2 3 ,.&.> x (e"_;e&^@];^;^@(] * ^.@]);(]^]);(^!._1)) y õıııııııÃıııııııÃııÃııÃııÃııÀ ş2.71828ş 1ş 1ş 1ş 1ş 1ş ş ş2.71828ş 4ş 1ş 1ş 4ş ş ş7.38906ş16ş 4ş 4ş12ş ş ş20.0855ş64ş27ş27ş24ş ÁıııııııÂıııııııÂııÂııÂııÂııã S2=. %.@S1=. (^!._1/~ %. ^/~) @ i. (S1;S2) 5 Signed Stirling numbers (transform õııııııııııııÃıııııııııÀ between ordinary and stope polynomials) ş1 0 0 0 0ş1 0 0 0 0ş ş0 1 _1 2 _6ş0 1 1 1 1ş ş0 0 1 _3 11ş0 0 1 3 7ş ş0 0 0 1 _6ş0 0 0 1 6ş ş0 0 0 0 1ş0 0 0 0 1ş ÁııııııııııııÂıııııııııã Natural Log ^. _ 0 0 Logarithm The natural logarithm (^.) is inverse to the exponential ^ (i.e., y=^.^y and y=^^.y). The base-x logarithm x^.y is the inverse of power (^) in the sense thaty = x^.x^y and y = x^x^.y. Certain properties of logarithms are illustrated below: ^. ^: _1 ^ x=. 4 [ y=. 0 1 2 3 (x^y);(x^.x^y);(x^.y);(x^x^.y) õıııııııııÃıııııııÃııııııııııııııııııÃıııııııÀ ş1 4 16 64ş0 1 2 3ş__ 0 0.5 0.7924813ş0 1 2 3ş ÁıııııııııÂıııııııÂııııııııııııııııııÂıııııııã logtable=. ^./~@i. <6.2 ": logtable 6 õııııııııııııııııııııııııııııııııııııÀ ş _. 0.00 0.00 0.00 0.00 0.00ş ş __ 0.00 _ _ _ _ş ş __ 0.00 1.00 1.58 2.00 2.32ş ş __ 0.00 0.63 1.00 1.26 1.46ş ş __ 0.00 0.50 0.79 1.00 1.16ş ş __ 0.00 0.43 0.68 0.86 1.00ş Áııııııııııııııııııııııııııııııııııııã The derivative of the natural logarithm is the reciprocal. For example: D=. ("0)(D.1) ^. D y=. 0 1 2 3 4 5 6 _ 1 0.5 0.333333 0.25 0.2 0.166667 % ^. D y 0 1 2 3 4 5 6 POWER u ^: n _ _ _ (see next topic) Two cases occur: a numeric integer n, and a gerund n. Numeric case. The verb u (or x&u) is applied n times. For example: (];+/\;+/\^:2;+/\^:0 1 2 3 _1 _2 _3 _4) y=. 1 2 3 4 5 õıııııııııÃıııııııııııÃııııııııııııÃıııııııııııııÀ ş1 2 3 4 5ş1 3 6 10 15ş1 4 10 20 35ş1 2 3 4 5ş ş ş ş ş1 3 6 10 15ş ş ş ş ş1 4 10 20 35ş ş ş ş ş1 5 15 35 70ş ş ş ş ş1 1 1 1 1ş ş ş ş ş1 0 0 0 0ş ş ş ş ş1 _1 0 0 0ş ş ş ş ş1 _2 1 0 0ş ÁıııııııııÂıııııııııııÂııııııııııııÂıııııııııııııã An infinite power n produces the limit of the application of u. For example, if x=. 2 and y=. 1, then x o.^:_ y is 0.73908, the solution of the equation y=Cos y. If n is negative, the obverse u^:_1 is applied |n times. The obverse (which is normally the inverse) is specified for five cases: 1. The pairs in the following lists: < <: + +. +: +~ - -. *. *: *~ % > >: + j./"1"_ -: -: - -. r./"1"_ %: %: % --------------------------------------------------------------------------------------------------------------------- %. ^ |. |: ,: ,~ %. ^. |. |: {. (<.@-:@#)&{. ------------------------------------------------------------------------------------------------------------------- ;: #. ". ;~ 3!:1 3!:3 ;@(,&' '&.>"1) #: ": >@{. 3!:2 3!:2 ----------------------------------------------------------------------------------------------------------------- /: \: [ ] o. C. j. p. r. /: /:@|. [ ] %&(o.1) C. %&0j1 p. %&0j1@^. 2. Obviously invertible monads such as -&3 and 10&^. and 1á0á2&|: and 3&|. and 1&o. and a.&i. as well as u@v and u&v and u&.v if u and v are invertible. 3. Monads of the form v/\ and v/\. where v is one of + * % = ~: 4. Obverses specified by :. 5. All others by a linear approximation Gerund case. (Compare with the gerund case of the adverb }) x u^:(v0`v1`v2)y (x v0 y)u^:(x v1 y) (x v2 y) x u^:( v1`v2)y x u^: ([` v1`v2) y u^:( v1`v2)y u^: ( v1 y) ( v2 y) POWER u ^: v _ _ _ (see previous) The case of ^: with a verb right argument is defined in terms of the noun right argument case (u ^: n) as follows: x u ^: v y is x u^: (x v y) y u ^: v y is u^: (v y) y For example: x=. 1 3 3 1 y=. 0 1 2 3 4 5 6 x p. y 1 8 27 64 125 216 343 x p. ^: (]>3:)"1 0 y 0 1 2 3 125 216 343 a=. _3 _2 _1 0 1 2 3 %: a 0j1.73205 0j1.41421 0j1 0 1 1.41421 1.73205 * a _1 _1 _1 0 1 1 1 %: ^: * " 0 a 9 4 1 0 1 1.41421 1.73205 *: a 9 4 1 0 1 4 9 The following monads are equivalent: g=. u ^: p ^: _ h=. 3 : 't=. y. while. p t do. t=. u t end.' (See the example of ^ T. _ in the definition of the Taylor series T.) u=. -&3 [. p=. 0&< (g"0 ; h"0) i. 10 õıııııııııııııııııııııııııÃıııııııııııııııııııııııııÀ ş0 _2 _1 0 _2 _1 0 _2 _1 0ş0 _2 _1 0 _2 _1 0 _2 _1 0ş ÁıııııııııııııııııııııııııÂıııııııııııııııııııııııııã Shape Of $ _ 1 _ Shape $ y yields the shape of y as defined in II A. For example, the shape of a 2-by-3 matrix is 2 3, and the shape of the scalar 3 is an empty list (whose shape is 0).The rank of an argument is given by #@$. For example: rank=. #@$ (rank 3) , (rank ,3)0 1 (rank 3 4),(rank i.2 3 4)1 3 The shape of x$y is x,siy where siy is the shape of an item of y. For example: y=.3 4$'abcdefghijkl' y ; 2 2$ yõııııÃııııÀşabcdşabcdşşefghşefghşşijklş şş şijklşş şabcdşÁııııÂııııãThis example shows ho w the result is formed from the items of y, the last1-cell (abcd) showing that the selection is cyclic. The fit conjunction ($!.f) provides fill specified by the items of f. Since x $ y uses items from y, it is sometimes useful to ravel the right argument, as in x $ ,y. For example (using the y defined above): 2 3 $ ,y abc def The fit conjunction is often useful for appending zeros or spaces. For example: 8 $!.0 (2 3 4) 2 3 4 0 0 0 0 0 ]z=. 8$!.' ' 'abc' abc |. z cba 2 5$!.(<'');: 'zero one two three four five six' õııııÃıııÃıııÃıııııÃııııÀ şzeroşoneştwoşthreeşfourş ÈııııÊıııÊıııÊıııııÊıııı³ şfiveşsixş ş ş ş ÁııııÂıııÂıııÂıııııÂııııã SELF-REFERENCE $: _ _ _ $: is a proxy that assumes the result of the phrase in which it occurs, the phrase being terminated on the left by a copula or by the completion of the sentence. For example: 1:`(] * $:@<:)@.* 5120In the foregoing expression, the agenda chooses the verb ] * $:@<: as long as the argument (reduced by one each time by the application of the decrement) remains non-zero. When the argument becomes zero, the result of the right argument of @. is zero, and the constant function 1: is chosen.If $:@ wer e omitted from the expression, it would execute once only as follows: 1:`(] * <:)@.* 520The inclusion of self-reference ensures that the entire function is re-executed after decrementing the argument. EVOKE m ~ _ (see next topic) If m is a proverb (that is, the name of a verb), then 'm'~y equals m y. If: m=. +/ and a=. 1 2 3 4, then 'm' ~ a is 10 (that is, máa). REFLEXIVE u ~ _ ru lu PASSIVE (see previous) u~ y is y u y. For example, ^~á3 is 27, and +/~ i. n is an addition table. ~ commutes or crosses connections to arguments: x u~ y y u x. Certain uses of the reflexive and passive are illustrated below: x=. 1 2 3 4 [ y=. 4 5 6 x (,.@[ ; ^/ ; ^/~ ; ^/~@[ ; ]) y õıÃıııııııııııııÃıııııııııııııÃıııııııııııÃıııııÀ ş1ş 1 1 1ş4 16 64 256ş1 1 1 1ş4 5 6ş ş2ş 16 32 64ş5 25 125 625ş2 4 8 16ş ş ş3ş 81 243 729ş6 36 216 1296ş3 9 27 81ş ş ş4ş256 1024 4096ş ş4 16 64 256ş ş ÁıÂıııııııııııııÂıııııııııııııÂıııııııııııÂıııııã into=. %~ (i. 6) % 5 0 0.2 0.4 0.6 0.8 1 5 into i. 6 0 0.2 0.4 0.6 0.8 1 from=. -~ (i.6) - 5 _5 _4 _3 _2 _1 0 5 from i.6 _5 _4 _3 _2 _1 0 (x %/ y);(x %~/ y);(x %/~ y) õııııııııııııııııııÃıııııııııııııııııııÃııııııııııııııııııÀ ş0.25 0.2 0.1666667ş 4 5 6ş4 2 1.33333 1ş ş 0.5 0.4 0.3333333ş 2 2.5 3ş5 2.5 1.66667 1.25ş ş0.75 0.6 0.5ş1.33333 1.66667 2ş6 3 2 1.5ş ş 1 0.8 0.6666667ş 1 1.25 1.5ş ş ÁııııııııııııııııııÂıııııııııııııııııııÂııııııııııııııııııã Nub ~. _ ~.y selects the nub of y, that is, all of its distinct items. For example: y=. 3 3 $ 'ABCABCDEF' y;(~.y);(~.3);($~.3)õıııÃıııÃıÃıÀşABCşABCş3ş1şşABCşDEFş ş şşDEFş ş ş şÁıııÂıııÂıÂıã More precisely, the nub is found by selecting the leading item, suppressing from the argument all items tolerantly equal to it, selecting the next remaining item, and so on. The fit conjunction applies to nub to specify the tolerance used. If f is a costly function, it may be quicker to evaluate f y by first evaluating f~. y (which yields all of the distinct results required), and then distributing them to their appropriate positions. The inner product with the self-classification table (produced by =) can be used to effect this distribution. For example: f=. *: f y=. 2 7 1 8 2 8 1 8 4 49 1 64 4 64 1 64 ,.&.>(~. ; f@~. ; = ; (f@~.(+/ .*)=) ; f)y õıÃııÃıııııııııııııııÃııÃııÀ ş2ş 4ş1 0 0 0 1 0 0 0ş 4ş 4ş ş7ş49ş0 1 0 0 0 0 0 0ş49ş49ş ş1ş 1ş0 0 1 0 0 0 1 0ş 1ş 1ş ş8ş64ş0 0 0 1 0 1 0 1ş64ş64ş ş ş ş ş 4ş 4ş ş ş ş ş64ş64ş ş ş ş ş 1ş 1ş ş ş ş ş64ş64ş ÁıÂııÂıııııııııııııııÂııÂııã NUB=. 1 : '(x.@~. +/ . * =)' Adverb *: NUB y 4 49 1 64 4 64 1 64 nubindex=. ~. i. ] (nubindex ; (nubindex { ~.)) y õıııııııııııııııÃıııııııııııııııÀ ş0 1 0 2 0 1 3 2ş8 1 8 2 8 1 7 2ş ÁıııııııııııııııÂıııııııııııııııã Nub Sieve ~: _ 0 0 Not Equal ~:y is the boolean list b such that b#y is the nub of y. For example: ~: 'Mississippi'1 1 1 0 0 0 0 0 1 0 0 x~:y is 1 if x is tolerantly unequal to y. See Equal (=). The fit conjunction may be used to specify tolerance, as in ~:!.t . The result of nub-sieve can be used to select the nub as follows: y=. 8 1 8 2 8 1 7 2 ~. y 8 1 2 7 ~: y 1 1 0 1 0 0 1 0 (~: y) # y 8 1 2 7 y #~ ~: y 8 1 2 7 The dyad ~: applies to any argument, but for booleans it is called exclusive-or. For example: d=. 0 1 d ~:/ d 0 1 1 0 Not-equal, not equal, and the dual of equal with respect to not, all agree as illustrated below. (~:/ ; -.@=/ ; =&.-."0/)~ d õıııÃıııÃıııÀ ş0 1ş0 1ş0 1ş ş1 0ş1 0ş1 0ş ÁıııÂıııÂıııã Magnitude | _ 0 0 Residue |y is %:y*+y. For example: | 6 _6 3j46 6 5 The familiar use of residue is in determining the remainder on dividing a non-negative integer by a positive. Thus: 3 | 0 1 2 3 4 5 6 70 1 2 0 1 2 0 1. The definition y-x*<. y % x+0=x extends the residue to a zero left argument, and to negative and fractional arguments. For example: x=. 3 2 1 0 _1 _2 _3 [ y=. 0 1 2 3 4 5 6 7 8 x (,.@[ ; |/ ; ]) y õııÃııııııııııııııııııııııııÃıııııııııııııııııÀ ş 3ş0 1 2 0 1 2 0 1 2ş0 1 2 3 4 5 6 7 8ş ş 2ş0 1 0 1 0 1 0 1 0ş ş ş 1ş0 0 0 0 0 0 0 0 0ş ş ş 0ş0 1 2 3 4 5 6 7 8ş ş ş_1ş0 0 0 0 0 0 0 0 0ş ş ş_2ş0 _1 0 _1 0 _1 0 _1 0ş ş ş_3ş0 _2 _1 0 _2 _1 0 _2 _1ş ş ÁııÂııııııııııııııııııııııııÂıııııııııııııııııã To produce a true zero for cases such as (%3)|(2%3) the residue is made tolerant as shown in the definition of res below: res=. f`g@.agenda"_ 0 0 agenda=. ([ = 0:) +. (<. = >.)@S S=. ] % [ + [ = 0: f=. ] - [ * <.@S [. g=. ] * [ = 0: 0.1 res 2.5 3.64 2 _1.6 0 0.04 0 0 (,. ; res/~ ; |/~) a=. 2 -~ i.5 õııÃııııııııııııÃııııııııııııÀ ş_2ş 0 _1 0 _1 0ş 0 _1 0 _1 0ş ş_1ş 0 0 0 0 0ş 0 0 0 0 0ş ş 0ş_2 _1 0 1 2ş_2 _1 0 1 2ş ş 1ş 0 0 0 0 0ş 0 0 0 0 0ş ş 2ş 0 1 0 1 0ş 0 1 0 1 0ş ÁııÂııııııııııııÂııııııııııııã The dyad | applies to complex numbers, as discussed in McDonnell [11]. Moreover, the fit conjunction may be applied to control the tolerance used. Reverse |. _ 1 _ Rotate (Shift) |. y reverses the order of the items of y. For example: |. t=. 'abcdefg'gfedcba The right shift is the dyadic case of |.!.f with the left argument _1. For example: |.!.'#' t#abcdef |.!.10 i.3 3 10 10 10 0 1 2 3 4 5 x|.y rotates successive axes of y by successive elements of x. Thus: 1 2 |. i. 3 5 7 8 9 5 612 13 14 10 11 2 3 4 0 1 The phrase x |.!.f y produces a shift: the items normally brought around by the cyclic rotation are replaced by f unless f is empty (0=#f), in which case they are replaced by the normal fill defined under {. : 2 _2 |.!.'#'"0 1 tcdefg####abcde y=. a.{~ (a. i. 'A') + i. 5 6 p=. (] ; 2&|. ; _2&|. ; 2&|."1 ; 2&(|.!.'*'"1)) y q=. (] ; |. ; |."1 ; |.!.'*'"1 ; (2: |. ])) y p ,&< q õııııııııııııııııııııııııııııııııııııÃııııııııııııııııııııııııııııııııııııÀ şõııııııÃııııııÃııııııÃııııııÃııııııÀşõııııııÃııııııÃııııııÃııııııÃııııııÀş şşABCDEFşMNOPQRşSTUVWXşCDEFABşCDEF**şşşABCDEFşYZ[\]^şFEDCBAş*ABCDEşMNOPQRşş şşGHIJKLşSTUVWXşYZ[\]^şIJKLGHşIJKL**şşşGHIJKLşSTUVWXşLKJIHGş*GHIJKşSTUVWXşş şşMNOPQRşYZ[\]^şABCDEFşOPQRMNşOPQR**şşşMNOPQRşMNOPQRşRQPONMş*MNOPQşYZ[\]^şş şşSTUVWXşABCDEFşGHIJKLşUVWXSTşUVWX**şşşSTUVWXşGHIJKLşXWVUTSş*STUVWşABCDEFşş şşYZ[\]^şGHIJKLşMNOPQRş[\]^YZş[\]^**şşşYZ[\]^şABCDEFş^]\[ZYş*YZ[\]şGHIJKLşş şÁııııııÂııııııÂııııııÂııııııÂııııııãşÁııııııÂııııııÂııııııÂııııııÂııııııãş ÁııııııııııııııııııııııııııııııııııııÂııııııııııııııııııııııııııııııııııııã 1 _2 |. !. '*' 3{. y **GHIJ **MNOP ****** Transpose |: _ 1 _ Transpose |: reverses the order of the axes of its argument. For example: (] ; |:) i. 3 4õıııııııııÃııııııÀş0 1 2 3ş0 4 8şş4 5 6 7ş1 5 9şş8 9 10 11ş2 6 10şş ş3 7 11şÁıııııııııÂııııııã x|:y moves axes x to the tail end. If x is boxed, the axes in each box are run together to produce a single axis: y=. 3 4$'abcdefghijkl' y;(1 0|:y);(0|:y);((<0 1)|:y)õııııÃıııÃıııÃıııÀşabcdşaeişaeişafkşşefghşbfjşbfjş şşijklşcgkşcgkş şş şdhlşdhlş şÁııııÂıııÂıııÂıııã For example: y=. a.{~ (a. i. 'a') + i. 2 3 4 z=. y;(2 1 |: y);((<2 1) |: y);(|: i. 4 5) z ,&< |:&.> z õııııııııııııııııııııııııÃıııııııııııııııııııııııııÀ şõııııÃıııÃıııÃıııııııııÀşõııÃııÃııÃııııııııııııııÀş şşabcdşaeişafkş0 5 10 15şşşamşamşamş 0 1 2 3 4şş şşefghşbfjşmrwş1 6 11 16şşşeqşbnşfrş 5 6 7 8 9şş şşijklşcgkş ş2 7 12 17şşşiuşcoşkwş10 11 12 13 14şş şş şdhlş ş3 8 13 18şşş şdpş ş15 16 17 18 19şş şşmnopş ş ş4 9 14 19şşşbnş ş ş şş şşqrstşmquş ş şşşfrşeqş ş şş şşuvwxşnrvş ş şşşjvşfrş ş şş şş şoswş ş şşş şgsş ş şş şş şptxş ş şşşcoşhtş ş şş şÁııııÂıııÂıııÂıııııııııãşşgsş ş ş şş ş şşkwşiuş ş şş ş şş şjvş ş şş ş şşdpşkwş ş şş ş şşhtşlxş ş şş ş şşlxş ş ş şş ş şÁııÂııÂııÂııııııııııııııãş ÁııııııııııııııııııııııııÂıııııııııııııııııııııııııã DETERMINANT u . v 2 _ _ DOT PROD The phrases -/ . * and +/ . * are the determinant and permanent of square matrix arguments. More generally, the phrase u . v is defined in terms of a recursive expansion by minors along the first column, as discussed below. For vectors and matrices, the phrase xá+/á.á*áy is equivalent to the dot, inner, or matrix product of math; other rank-0 verbs such as <. and *. are treated analogously. In general, u . v is defined by u@(v"(1+lv,_)), restated in English below.