View source for System of subvarieties

''system of submanifolds''

Given a [[Differentiable manifold|differentiable manifold]] (or [[Algebraic variety|algebraic variety]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s0919701.png" />, a system of submanifolds (system of subvarieties) is a differentiably (algebraically) parametrized family of submanifolds (subvarieties) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s0919702.png" />. More generally, the notion makes sense and is useful also for other geometric objects, such as complex spaces, topological manifolds, algebras, etc. Such systems occur in may parts of mathematics both as tools and as objects to be investigated. Often additional requirements are added.

==1. Foliations.==
A codimension-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s0919704.png" /> foliation of a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s0919705.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s0919706.png" />-dimensional system of submanifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s0919707.png" /> such that locally near every point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s0919708.png" /> the system looks like the system of parallel translates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s0919709.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197010.png" />. Cf. [[Foliation|Foliation]] for more details.

==2. Nets.==
A net in differential geometry is a family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197011.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197012.png" />-parameter systems of smooth curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197013.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197014.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197015.png" /> such that through each point there passes precisely one curve from each family and such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197016.png" /> curves passing through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197017.png" /> intersect transversally (i.e. their tangent vectors at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197018.png" /> span the whole tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197019.png" />). Cf. [[Net (in differential geometry)|Net (in differential geometry)]]. Thus, a net consists so to speak of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197020.png" /> transversally intersecting codimension-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197021.png" /> foliations.

==3. Webs.==
More generally one can consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197022.png" /> "transversally intersecting" foliations of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197023.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197024.png" />-dimensional manifold for more general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197027.png" />. This leads to the theory of webs, cf. [[Webs, geometry of|Webs, geometry of]] and [[#References|[a13]]].

==4. Linear systems.==
When the subvarieties or submanifolds are given by equations, e.g. in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197028.png" /> or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197029.png" />, it makes sense to talk about linear systems. A linear system of subvarieties is given by a family of equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197031.png" /> are functions on the ambient manifold. Cf. also [[Linear system|Linear system]]. The notion of a linear algebraic system has received a great deal of attention particularly in analytic, projective and algebraic geometry, e.g. in the form of a (linear) system of subvarieties of a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197032.png" /> "cut out" by the system (*). Often the varieties in a linear system are all of the same dimension, or even of the same kind (apart from non-generic degeneracies).

==5. Web (in algebraic geometry).==
A three- (or higher-) parameter linear system of varieties, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197033.png" /> in (*). Sometimes also called a net.

==6. Net (in algebraic geometry).==
A two-parameter linear system of varieties, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197034.png" /> in (*). Cf. [[Net|Net]] for more details.

==7. Pencil (in geometry).==
By far the most attention has been paid to one-dimensional systems of varieties and manifolds. These are called pencils. Thus, the family of curves in the plane determined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197035.png" /></td> </tr></table>

for all possible values of the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197037.png" /> (except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197039.png" />) is a pencil (in fact, the pencil depends on the single parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197040.png" />). The equation of a pencil of surfaces in space is written similarly. The two equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197042.png" /> give two elements of the pencil (two curves or two surfaces) which determine the whole pencil. Every two elements of the pencil intersect in the same set of points — the support. The support of the pencil can contain real as well as imaginary points. If the original curves of the pencil are algebraic curves of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197044.png" />, then the support consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197045.png" /> points (real or imaginary, ordinary or exceptional, cf. [[Bezout theorem|Bezout theorem]]).

A pencil of straight lines is the set of all straight lines lying in a plane and passing through a fixed point (an ordinary pencil) or parallel to a fixed straight line (an exceptional pencil). The equation of a pencil of straight lines has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197046.png" /></td> </tr></table>

A pencil of planes is the set of all planes passing through a fixed straight line (an ordinary pencil) or parallel to a certain fixed plane (an exceptional pencil). The equation of a pencil of planes has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197047.png" /></td> </tr></table>

A pencil of circles is a one-parameter family of circles which linearly depends on the parameter. A pencil of circles contains circles and one straight line. The support of an (ordinary) pencil of circles consists of the two circular points and two ordinary points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197049.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197050.png" />, then a pencil of circles can be defined as the set of circles (taking straight lines as circles of infinite radius) which pass through the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197052.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197053.png" /> it is necessary to require in addition that the circles all touch at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197054.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197056.png" /> are real and distinct, the pencil of circles is said to be elliptic (Fig. a), if they coincide (and are real) it is called parabolic (Fig. b) and if they are imaginary (and distinct) it is said to be hyperbolic (Fig. c). A set of concentric circles (Fig. d) is an exceptional pencil.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091970a.gif" />

Figure: s091970a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091970b.gif" />

Figure: s091970b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091970c.gif" />

Figure: s091970c

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091970d.gif" />

Figure: s091970d

Every ordinary pencil of circles has a so-called [[Radical axis|radical axis]] — a straight line each point of which has the same power (different for different points) with respect to all circles of the pencil (cf. [[Degree of a point|Degree of a point]]). The radical axis of an elliptic pencil passes through the common points of the circles; for a parabolic pencil it is their common tangent; for a hyperbolic pencil it is the line joining the centres of the two circles which are orthogonal to all the circles of the pencil. The centres of the circles of a pencil lie on a straight line which is perpendicular to the radical axis. The point of intersection of the line of the centres of a pencil and its radical axis is called the centre of the pencil. The power of the centre of the pencil with respect to any circle of the pencil is the same and is called the power of the pencil. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197057.png" />-axis is the line of centres of the circles of the pencil, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197058.png" />-axis is the radical axis of the pencil, then the equation of an arbitrary circle of the pencil has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197059.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197060.png" /> is the parameter determining a given circle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197061.png" /> is the power of the pencil. For an elliptic pencil <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197062.png" />, for a parabolic pencil <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197063.png" />, for a hyperbolic pencil <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197064.png" /> (the power of an exceptional pencil can be considered to be infinite).

The circles which are orthogonal to all the circles of a given pencil form a pencil themselves; this pencil is said to be conjugate to the given pencil. An elliptic pencil is conjugate to a hyperbolic pencil and a parabolic pencil is conjugate to a parabolic pencil.

Any pencil of circles is the intersection of two nets (cf. [[Net|Net]]) of circles.

A pencil of spheres is a one-parameter family of spheres which linearly depends on the parameter. Any two spheres of the pencil intersect in a circle of real, zero or imaginary radius. In the first case the pencil of spheres is said to be elliptic, it consists of all spheres passing through a given circle; in the second case it is parabolic, and the pencil consists of all the spheres touching at a common point; in the third case it is hyperbolic, and the pencil consists of all the spheres which are orthogonal to three given spheres which intersect at two points. A pencil of spheres has a so-called radical plane, each point of which has the same power (different for different points) with respect to all the spheres of the pencil; the centres of all the spheres of a pencil lie on a single straight line, which is perpendicular to the radical plane.

A pencil of spheres is the intersection of three nets of spheres with centres lying on a single straight line.

In projective geometry an algebraic pencil of lines is the set of all lines in the projective plane with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197065.png" /> satisfying the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197066.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197067.png" /> is a polynomial not identically zero and homogeneous with respect to the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197070.png" />; the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197071.png" /> is called the degree (or order) of the pencil of lines. A first-order algebraic pencil of lines is given by an equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197072.png" /></td> </tr></table>

and is the set of all lines passing through the point with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197073.png" />.

A second-order algebraic pencil of lines is given by an equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197074.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197075.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197076.png" /> are real numbers at least one of which is non-zero. If the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197078.png" />, is non-zero, the second-order pencil of lines is called non-degenerate, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197079.png" /> it is degenerate. Each non-degenerate second-order pencil is the set of tangents to a non-degenerate [[Second-order curve|second-order curve]]; each non-degenerate second-order curve is the [[Envelope|envelope]] of a certain non-degenerate pencil of the second order.

More assorted algebraic and geometric pencils are discussed in [[Lobachevskii geometry|Lobachevskii geometry]]; [[Darboux quadric|Darboux quadric]]; [[Halphen pencil|Halphen pencil]]; [[Cubic|Cubic]]; [[Monodromy transformation|Monodromy transformation]]; [[Circle|Circle]].

Note that in some of the literature dealing with webs, nets and pencils in the sense of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197080.png" />-, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197081.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197082.png" />-dimensional families one occasionally adds the requirement that through each point there pass precisely a given number of members of the system, while the linearity condition (*) is dropped. Cf. e.g. [[#References|[a14]]].

==8. Fibrations.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197083.png" /> be a system of submanifolds or subvarieties such that through each of the points of the ambient manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197084.png" /> there passes exactly one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197085.png" />. Then there results a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197086.png" />. A priori, this mapping need not even be continuous. But if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197087.png" /> has appropriate properties (e.g. differentiable, holomorphic, algebraic, depending on the kind of objects under consideration), one obtains a fibration. Conversely, a surjective mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197088.png" /> defines a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197089.png" />. If the fibration is locally trivial, i.e. if locally in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197090.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197091.png" /> looks like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197092.png" />, one usually speaks of a fibre bundle instead of a fibration.

==9. Pencils of matrices.==
The word "pencil" is also used in other fields of mathematics, to denote one-parameter (linear) families of objects. Thus, one speaks of pencils of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197093.png" />, and especially of pencils of matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197094.png" />, or, inhomogeneously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197095.png" />. The pencil <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197096.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197098.png" /> have coefficients in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s09197099.png" />, can also be regarded as a matrix over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970100.png" />.

Two polynomial matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970102.png" />, i.e. matrices with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970103.png" />, are equivalent if there are invertible matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970105.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970106.png" />. (Note that a square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970107.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970108.png" /> is invertible if and only if its [[Determinant|determinant]] is a non-zero element of the field of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970109.png" />.) The two polynomial matrices are strictly equivalent if there are invertible matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970111.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970112.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970113.png" />. Equivalence of matrices is described precisely by the Smith canonical form, cf. [[Normal form|Normal form]] for more details.

Consider a pencil of square matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970114.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970115.png" />. For such pencils strict equivalence is the same as equivalence. A pencil of matrices is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970116.png" /> does not vanish identically. All other pencils are called singular.

The theory of invariants and canonical form for pencils of matrices was developed by L. Kronecker [[#References|[a10]]]; cf. also [[#References|[a9]]] for a complete account and some applications.

In (automatic) linear control theory one studies linear control systems

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970117.png" /></td> </tr></table>

determined by a pair of matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970118.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970119.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970120.png" />) of sizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970122.png" />, respectively. The so-called feedback group is generated by the following transformations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970123.png" /></td> </tr></table>

(this corresponds to a base change in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970124.png" /> of controls);

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970125.png" /></td> </tr></table>

(this corresponds to a base change in the state space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970126.png" />);

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970127.png" /></td> </tr></table>

(this corresponds to state feedback: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970128.png" />).

Associate to the linear control system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970129.png" /> the singular pencil

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970130.png" /></td> <td valign="top" style="width:5%;text-align:right;">(**)</td></tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970131.png" /> matrices. Now observe, [[#References|[a2]]], that two pencils of the form (**) are strictly equivalent if and only if the corresponding control systems are equivalent under the feedback group.

The control system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970132.png" /> is completely reachable if the columns of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970134.png" /> span all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970135.png" />. For pencils of the form (**) of completely-reachable control systems, many of the Kronecker invariants for pencils vanish; the most important that remain are a finite number of integer-valued ones. These can be described as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970136.png" /> be the dimension of the space spanned by the columns of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970138.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970139.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970140.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970141.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970142.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970143.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970144.png" />. Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970145.png" /> be the dual partition of the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970146.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970147.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970148.png" />. (Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970149.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970150.png" />, because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970151.png" />.) These <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970152.png" /> are potentially non-zero Kronecker invariants of (**). They are called the Kronecker indices in control theory. There is a corresponding state-feedback canonical form for a completely-reachable pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970153.png" /> in which the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970154.png" /> is a [[Jordan matrix|Jordan matrix]] with Jordan blocks <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970156.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970157.png" /> consists of standard unit vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970159.png" />, and, possibly, other vectors. This canonical form is called the Brunovsky canonical form. See [[#References|[a12]]] and, e.g., [[#References|[a11]]] for related matters.

==10. Deformations (in algebra, algebraic, analytic and differential geometry, etc.).==
Still another way of looking at a family of (sub) manifolds or varieties (or other structures) is as a [[Deformation|deformation]].

==Note on terminology.==
The terminology "web" , "net" , "pencil" as in 5.–7. above for, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970160.png" />-, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970161.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091970/s091970162.png" />-dimensional (linear) systems is not completely standard. In particular, one also finds "net" instead of "web" , and sometimes "bundle" in the meaning of both "net" (as in 6.) and "pencil" (as in 7. or, rarely, as in 9.).

The material in 7. above is basically a translation of the subsection on pencils by A.B. Ivanov appearing under "puchok" in the original Russian version of the Encyclopaedia. The remainder of this article is original to this annotated and expanded translation.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.M. Postnikov, "Analytic geometry" , Moscow (1973) (In Russian) {{MR|0987205}} {{MR|0890748}} {{ZBL|0662.51001}} {{ZBL|0482.51001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.L. Coolidge, "A treatise on the circle and the sphere" , Chelsea, reprint (1971) {{MR|0389515}} {{ZBL|0251.50002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S.-T. Hu, "Differentiable manifolds" , Holt, Rinehart &amp; Winston (1969) {{MR|0245030}} {{ZBL|0182.26102}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Morrow, K. Kodaira, "Complex manifolds" , Holt, Rinehart &amp; Winston (1971) {{MR|0302937}} {{ZBL|0325.32001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Berger, "Geometry" , '''1–2''' , Springer (1987) (Translated from French) {{MR|0903026}} {{MR|0895392}} {{MR|0882916}} {{MR|0882541}} {{ZBL|0619.53001}} {{ZBL|0606.51001}} {{ZBL|0606.00020}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J. Coolidge, "A history of the conic sections and quadratic surfaces" , Dover, reprint (1968) {{MR|}} {{ZBL|0202.00103}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) {{MR|1531486}} {{MR|0123930}} {{ZBL|0095.34502}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , '''2''' , Chelsea, reprint (1959) pp. Chapt. XII (Translated from Russian) {{MR|0107649}} {{MR|0107648}} {{ZBL|0085.01001}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> L. Kronecker, "Algebraische Reduktion der Schaaren bilinearer Formen" ''Sitzungsber. Akad. Berlin'' (1890) pp. 763–776</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> M. Hazewinkel, "A partial survey of the uses of algebraic geometry in system and control theory" , ''Symp. Math. INDAM (Rome 1979)'' , '''24''' , Acad. Press (1981) pp. 245–292 {{MR|0619251}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> R.E. Kalman, "Kronecker invariants and feedback" L. Weiss (ed.) , ''Ordinary Differential Equations'' , Acad. Press (1972) pp. 459–471 {{MR|0421751}} {{ZBL|0308.93008}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> V.V. Goldberg, "Theory of multicodimensional webs" , Kluwer (1988) {{MR|}} {{ZBL|0668.53001}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> F. Severi, "Vorlesungen über algebraische Geometrie" , Johnson, reprint (1968) {{MR|0245574}} {{ZBL|48.0687.01}} </TD></TR></table>