In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a shear matrix or transvection is an
elementary matrix In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multip ...
that represents the
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
of a multiple of one row or column to another. Such a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
may be derived by taking the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
and replacing one of the zero elements with a non-zero value.
The name ''shear'' reflects the fact that the matrix represents a
shear transformation. Geometrically, such a transformation takes pairs of points in a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
that are purely axially separated along the axis whose row in the matrix contains the shear element, and effectively replaces those pairs by pairs whose separation is no longer purely axial but has two vector components. Thus, the shear axis is always an
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of ''S''.
Definition
A typical shear matrix is of the form
This matrix shears parallel to the ''x'' axis in the direction of the fourth dimension of the underlying vector space.
A shear parallel to the ''x'' axis results in
and
. In matrix form:
Similarly, a shear parallel to the ''y'' axis has
and
. In matrix form:
In 3D space this matrix shear the YZ plane into the diagonal plane passing through these 3 points:
The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
will always be 1, as no matter where the shear element is placed, it will be a member of a skew-diagonal that also contains zero elements (as all skew-diagonals have length at least two) hence its product will remain zero and will not contribute to the determinant. Thus every shear matrix has an
inverse, and the inverse is simply a shear matrix with the shear element negated, representing a shear transformation in the opposite direction. In fact, this is part of an easily derived more general result: if ''S'' is a shear matrix with shear element
, then ''S
n'' is a shear matrix whose shear element is simply ''n''
. Hence, raising a shear matrix to a power ''n'' multiplies its
shear factor by ''n''.
Properties
If ''S'' is an ''n'' × ''n'' shear matrix, then:
* ''S'' has
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
''n'' and therefore is invertible
* 1 is the only
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of ''S'', so det ''S'' = 1 and trace ''S'' = ''n''
* the
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of ''S'' (associated with the eigenvalue 1) has ''n''−1 dimensions.
* ''S'' is
defective
Defective may refer to::
*Defective matrix, in algebra
*Defective verb, in linguistics
*Defective, or ''haser'', in Hebrew orthography, a spelling variant that does not include mater lectionis
*Something presenting an anomaly, such as a product de ...
* ''S'' is asymmetric
* ''S'' may be made into a
block matrix by at most 1 column interchange and 1 row interchange operation
* the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
,
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
, or any higher order interior capacity of a
polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
is invariant under the shear transformation of the polytope's vertices.
Composition
Two or more shear transformations can be combined.
If two shear matrices are
and
then their composition matrix is
which also has determinant 1, so that area is preserved.
In particular, if
, we have
which is a
positive definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a co ...
.
Applications
* Shear matrices are often used in
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
.
Computer Graphics
Apueva A. Desai, pp. 162-164
See also
*Transformation matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then
T( \mathbf x ) = A \mathbf x
for some m \times n matrix ...
Notes
References
*
{{DEFAULTSORT:Shear Matrix
Matrices
Linear algebra
Sparse matrices