Fact-checked by Grok 2 weeks ago

Similarity transformation

A similarity transformation, also known as a similitude, is a geometric mapping in the Euclidean plane or space that preserves angles and scales all distances by a fixed positive ratio k > 0, thereby transforming figures into similar figures that have the same shape but possibly different sizes.^[1] These transformations are conformal, meaning they maintain the local shape around points, and are fundamental in similarity geometry, which studies properties invariant under such scalings and rigid motions.^[2] Similarity transformations can be decomposed into a composition of an isometry (such as translation, rotation, or reflection) and a homothety (dilation centered at a point), allowing them to enlarge, reduce, or rigidly move objects while preserving relative proportions.^[3] They are classified into two types: direct similarities, which preserve orientation (e.g., via the complex mapping z' = a z + b with a \neq 0), and opposite similarities, which reverse orientation (e.g., via reflection combined with dilation and translation, as in z' = \bar{a} \bar{z} + b).^[3] Key properties include mapping parallel lines to parallel lines, perpendicular lines to perpendicular lines, and circles to circles, making them essential for theorems on similar triangles and proportionality in geometry.^[3] In linear algebra, a similarity transformation refers to a change of basis for matrices, where a matrix A is transformed to A' = P^{-1} A P for some invertible matrix P, preserving algebraic invariants such as eigenvalues, trace, determinant, and characteristic polynomial.^[1] This concept connects to diagonalization, where a matrix is similar to a diagonal form if it has a full set of eigenvectors, and extends to applications in group theory and fractal geometry via iterated function systems.^[1]

Geometric transformations

Definition and basic properties

In geometry, a similarity transformation is a bijection of the Euclidean plane that scales all distances between points by a fixed positive constant factor k > 0, while preserving the overall shape of figures.^[3] Formally, for any two points A and B, the distance between their images S(A) and S(B) satisfies |S(A)S(B)| = k \cdot |AB|.^[4] Such transformations can be expressed as a composition of isometries—rigid motions like translations, rotations, or reflections—and a dilation (or homothety), which uniformly scales distances from a fixed center point.^[5] Similarity transformations exhibit several fundamental properties that distinguish them from more general mappings. They preserve collinearity, mapping collinear points to collinear points, and parallelism, sending parallel lines to parallel lines.^[3] Additionally, they maintain angles between lines or curves, ensuring that congruent angles in the original figure correspond to congruent angles in the image, though the orientation of these angles may depend on the type of similarity.^[6] Unlike isometries (where k = 1), similarities do not preserve absolute distances but do preserve ratios of distances within a figure, such as the ratios of sides in polygons or segments along a line.^[5] The scale factor k determines the extent of enlargement or reduction effected by the transformation. If k > 1, the transformation enlarges figures, increasing all distances by the factor k; for example, a dilation with center O and k = 2 maps a segment AB to a segment twice as long, with O as the midpoint if A and B are equidistant from O.^[5] Conversely, if $0 < k < 1, it reduces figures proportionally, as in a contraction that halves all distances. When k = 1, the similarity reduces to an isometry, preserving both shapes and sizes.^[4] Similarities are classified based on their effect on orientation. Direct similarities preserve the orientation of figures, mapping clockwise turns to clockwise turns and counterclockwise to counterclockwise; they arise from compositions of translations, rotations, and dilations.^[3] Opposite similarities, in contrast, reverse orientation, turning clockwise to counterclockwise, and incorporate a reflection alongside dilations and other rigid motions.^[3]

Composition and classification

Similarity transformations form a group under composition, as the composition of any two similarities is again a similarity, the identity transformation is a similarity with scale factor 1, and every similarity has an inverse that is also a similarity. The scale factor of the composition of two similarities with scale factors k_1 and k_2 is the product k_1 k_2, ensuring that distances are scaled consistently. For instance, composing two dilations centered at the same point with scales k_1 and k_2 results in a single dilation with scale k_1 k_2; if one or both include translations, the overall effect combines scaling, rotation (if present), and a net translation. Rigid motions, which are isometries with scale factor 1, compose to yield similarities that are also isometries, preserving both shape and size. Similarities are classified based on their effect on orientation and their geometric components. Direct similarities preserve orientation and consist of an orientation-preserving isometry (such as translation or rotation) composed with a dilation; they map figures to their non-reflected counterparts while scaling distances. Opposite similarities reverse orientation and include a reflection or glide reflection combined with dilation, effectively incorporating a mirroring effect alongside scaling.^[3] A special subclass is spiral similarities, which are direct similarities comprising a rotation and a dilation both centered at the same fixed point, often used to map one line segment to another while preserving angles between them. Fixed points play a key role in characterizing similarities. A pure dilation has a unique fixed point at its center, where points remain unchanged while others are scaled radially. When translation is involved, such as in a similarity combining dilation and translation, there is generally exactly one fixed point, which serves as the center of similitude balancing the scaling and displacement; this point can be constructed geometrically by intersecting lines connecting corresponding vertices of similar figures. For example, in aligning two similar rectangles scaled by a factor of 1/2, the fixed point is found at the intersection of lines joining midpoints of corresponding sides, ensuring the transformation maps one to the other without fixed points at infinity unless the scale is 1. Similarity transformations generalize congruences, which are the special case where the scale factor k = 1, reducing similarities to isometries that preserve distances exactly. To illustrate composition step-by-step, consider mapping a triangle ABC to a similar triangle A'B'C' with k = 2: first apply a dilation centered at A with scale 2 to get A''B''C'' (fixing A while doubling sides from it), then compose with a rotation around A'' to align B'' to A'B', followed by a translation to position A'' at A'; the result is a direct similarity with overall scale 2, demonstrating how basic components build complex mappings while maintaining shape.

Applications in geometry

In geometry, two figures are similar if one can be obtained from the other through a similarity transformation, which preserves angles and proportionality of distances.^[7] This concept is foundational for understanding relationships between shapes, particularly in the case of triangles, where similarity criteria such as the Angle-Angle (AA) and Side-Angle-Side (SAS) theorems can be derived by applying similarity transformations to map one triangle onto another while maintaining corresponding angles and proportional sides.^[8] For instance, the AA theorem holds because a similarity transformation that aligns two angles of the triangles automatically scales the third angle to match due to the sum of angles in a triangle being 180 degrees.^[9] Similarity transformations provide a powerful tool for proofs, particularly in establishing proportionality among similar polygons. By composing translations, rotations, reflections, and dilations, one can transform one polygon to overlay another, demonstrating that corresponding sides are in constant ratio and angles are equal.^[10] For example, to prove the side ratios in two similar triangles, a similarity transformation can map the vertices of the first triangle to the second, showing that the scaling factor k applies uniformly to all sides, thus justifying theorems like SAS similarity where proportional sides enclosing equal angles imply overall similarity.^[9] This transformational approach avoids direct measurement and relies on the invariance of angles under similarities. Practical applications of similarity transformations abound in various geometric contexts. In fractals, self-similarity arises through iterative applications of dilations, where each subscale is a scaled copy of the whole, as seen in the construction of the Sierpinski triangle by repeatedly applying a dilation with scale factor 1/2 centered at the vertices.^[11] Tessellations can incorporate similarity transformations to create patterns with scaled variants of a base tile, allowing for hierarchical designs that blend repetition with enlargement, such as in decorative motifs where smaller tiles mirror larger ones proportionally.^[12] In computer graphics, similarity transformations are essential for scaling sprites while preserving their shape and orientation, enabling efficient rendering of resized objects in video games and animations through combined rotation, translation, and uniform scaling.^[13] A specific example of a dilation, a key component of similarity transformations, in coordinate geometry involves mapping a point (x, y) relative to a center (h, v) by a scale factor k. The transformed coordinates are given by:

(x', y') = \left( k(x - h) + h, \, k(y - v) + v \right)

This formula enlarges or reduces the figure about the center (h, v); for instance, with center (0, 0) and k = 2, the point (1, 1) maps to (2, 2), doubling distances from the origin while preserving directions.^[14]

Matrix transformations

Definition and equivalence

In linear algebra, a similarity transformation relates two square matrices A and B of the same dimension n \times n through an invertible matrix P, such that

B = P^{-1} A P.

This definition captures the essence of representing the same linear operator on \mathbb{R}^n (or a general vector space) with respect to different bases, where the columns of P form the new basis vectors.^[15]^[16] The similarity relation is an equivalence relation on the set of n \times n matrices, as it satisfies reflexivity (A \sim A via P = I, the identity matrix), symmetry (if B = P^{-1} A P, then A = P B P^{-1}), and transitivity (if A \sim B and B \sim C, then A \sim C via the product of the corresponding invertible matrices).^[17]^[16] Consequently, similarity partitions matrices into equivalence classes, where each class groups all matrix representations of a given linear transformation under varying choices of basis.^[16] Geometrically, similarity embodies a change of coordinates that preserves the intrinsic linear action of the transformation while altering its matrix representation, equivalent to rotating, scaling, or shearing the basis to view the operator in a new frame.^[15] For example, if B describes a rotation by an angle \theta in the standard basis, then A = P B P^{-1} describes the identical rotation in a basis spanned by the columns of P, which may visually distort familiar shapes—such as mapping a circle to an ellipse—without changing the underlying transformation.^[15]^[18] A key special case of similarity is diagonalization: a square matrix A is diagonalizable if it is similar to a diagonal matrix D, meaning there exists an invertible P such that P^{-1} A P = D, where the columns of P are eigenvectors of A and the diagonal entries of D are the corresponding eigenvalues.^[19] This simplifies computations involving powers of A, as A^k = P D^k P^{-1}, leveraging the ease of raising diagonal matrices to powers.^[19]

Properties and invariants

Similar matrices A and B = P^{-1}AP over a field (typically \mathbb{R} or \mathbb{C}) preserve fundamental algebraic properties, serving as invariants that fully characterize their equivalence class under similarity. These invariants include the trace, which is the sum of the diagonal entries and equals the sum of eigenvalues (counting multiplicities); the determinant, which is the product of the eigenvalues; the characteristic polynomial \det(\lambda I - A), a monic polynomial of degree n whose roots are the eigenvalues; the eigenvalues themselves with algebraic multiplicities; and the minimal polynomial, the monic polynomial of least degree that annihilates A.^[20]^[21] The rank of A, defined as the dimension of its image, and the nullity, the dimension of its kernel, are also preserved under similarity, as these are intrinsic properties of the associated linear transformation independent of the basis.^[20] Furthermore, over algebraically closed fields like \mathbb{C}, every matrix is similar to a unique Jordan canonical form consisting of Jordan blocks, up to permutation of the blocks; this form is determined solely by the eigenvalues and the sizes of the blocks, which correspond to the geometric multiplicities and the structure of the generalized eigenspaces.^[21]^[20] While these invariants remain unchanged, certain features transform under similarity: specifically, eigenvectors of A are not generally eigenvectors of B, though the corresponding eigenspaces are isomorphic as vector spaces, preserving their dimensions (the geometric multiplicities).^[21] A key example of invariance is the trace: \operatorname{tr}(P^{-1}AP) = \operatorname{tr}(AP P^{-1}) by the cyclic property of the trace, which equals \operatorname{tr}(A), ensuring the sum of eigenvalues is basis-independent.^[20]

Computational aspects and applications

Computing similarity transformations for matrices involves algorithms that exploit the structure-preserving properties of these transformations to simplify matrix analysis. The QR algorithm, an iterative method based on successive QR decompositions, is widely used to compute the Schur decomposition of a square matrix A, yielding A = Q T Q^H where Q is unitary and T is upper triangular; this represents a similarity transformation that triangularizes A while preserving its eigenvalues on the diagonal of T.^[22] For matrices that are diagonalizable, the transformation matrix P can be constructed from the eigenvectors of A, such that P^{-1} A P = D where D is diagonal, enabling efficient computation of functions like powers or exponentials of A.^[23] Numerical challenges arise in these computations, particularly when the transformation matrix P is ill-conditioned, as small perturbations in A can lead to large errors in the computed eigenvectors and thus in P, amplifying rounding errors in floating-point arithmetic.^[24] The singular value decomposition (SVD), which factorizes A = U \Sigma V^H with unitary U and V and diagonal \Sigma, serves as a related but distinct tool; unlike similarity transformations that require invertibility and preserve eigenvalues exactly, SVD handles rectangular or rank-deficient matrices and provides a stable approximation for low-rank structures, though it does not yield a similar matrix unless A is normal.^[25] In applications, similarity transformations via the Jordan canonical form facilitate solving systems of ordinary differential equations (ODEs) of the form \mathbf{x}' = A \mathbf{x}; by finding an invertible P such that P^{-1} A P = J where J is the Jordan form, the system decouples into simpler equations in the transformed coordinates \mathbf{y} = P^{-1} \mathbf{x}, allowing explicit solutions involving exponentials of Jordan blocks.^[26] Principal component analysis (PCA) employs orthogonal similarity transformations on the covariance matrix C of centered data, diagonalizing it as Q^T C Q = \Lambda where Q consists of eigenvectors (principal components) and \Lambda is diagonal with variances; this reduces dimensionality by retaining components corresponding to large eigenvalues.^[27] In quantum mechanics, unitary similarity transformations U^\dagger O U (with U unitary) preserve the spectrum of Hermitian observables O, enabling basis changes that simplify calculations of expectation values or time evolution while maintaining physical interpretability.^[28] A representative example is the 2D rotation matrix R = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}, which can be diagonalized over the complex numbers via a similarity transformation P^{-1} R P = D = \begin{pmatrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{pmatrix}, where P has columns as eigenvectors; this form simplifies computations such as powers R^n = P D^n P^{-1}, avoiding repeated trigonometric evaluations.^[23]

Historical development

Early concepts in geometry

The concept of similarity in geometry traces its origins to ancient Greece, where Euclid's Elements (c. 300 BCE) laid foundational ideas through proportion theorems applied to triangles and polygons, without explicitly framing them as transformations. In Book VI, Euclid defined similar figures as those with equal corresponding angles and proportional sides, using propositions like VI.4 to establish that equiangular triangles have sides proportional to one another, building on earlier work by Eudoxus on ratios for incommensurable magnitudes. These ideas implicitly relied on scaling relations to prove area proportions and parallel line intercepts (e.g., VI.2), forming the basis for understanding shape preservation under uniform enlargement or reduction, though Euclid focused on static proportions rather than dynamic mappings.^[29]^[30] During the Renaissance, notions of similarity emerged in artistic applications of perspective, particularly through Filippo Brunelleschi's experiments around 1420, which employed similar triangles to model scaling effects for depth illusion on flat surfaces. Brunelleschi demonstrated how distant objects appear diminished in size via proportional relations between observer distance and apparent height, using mirrors and peepholes to verify geometric accuracy in depictions of Florence's Baptistery. This practical linkage of scaling to visual projection anticipated formal geometric transformations, emphasizing similarity in preserving angular relations while altering magnitudes, though it remained tied to empirical drawing techniques rather than abstract theory.^[31] In the 19th century, similarity concepts gained formal distinction within projective geometry through Jean-Victor Poncelet's Traité des propriétés projectives des figures (1822), which identified projective invariants like the cross-ratio while separating similarities—transformations preserving angles and shape ratios—from broader projections that distort metrics. Poncelet emphasized properties invariant under perspective mappings, contrasting them with Euclidean similarities that maintain proportional distances. Complementing this, Michel Chasles introduced the term "homothety" (or dilation) in his geometric works around 1837, formalizing central scalings as transformations with a fixed point and ratio that generate similar figures, thus bridging projective and metric geometries.^[32]^[33] A pivotal milestone came with Felix Klein's Erlangen Program (1872), which classified geometries by their underlying transformation groups, positioning similarities as the group preserving angles, parallelism, and shape for Euclidean spaces, extending beyond rigid isometries to include scalings. Klein's framework unified prior developments by viewing similarity geometry as subordinate to projective geometry, with homotheties as key elements defining invariants like ratios of lengths along lines. This group-theoretic perspective solidified similarities as a core structure in classical geometry.^[34]

Evolution in linear algebra

The concept of similarity transformations emerged in the 19th century as part of the foundational development of matrix theory, largely through Arthur Cayley's pioneering work. In his 1858 memoir, Cayley formalized matrices as representations of linear transformations between vector spaces of the same dimension, defining operations such as addition, multiplication, and inversion that align with the composition of these transformations.^[35] This framework implicitly introduced the notion of matrix equivalence, where two matrices are considered equivalent if one can be obtained from the other via invertible transformations, laying the groundwork for similarity as a special case where the transforming matrices are the same on both sides (i.e., B = P^{-1} A P).^[36] Cayley's emphasis on invariants under such transformations—properties preserved across equivalent representations—provided an algebraic lens for understanding structural similarities in linear maps, influencing subsequent classifications.^[37] A significant advancement came in 1870 with Camille Jordan's introduction of the canonical form that bears his name, which explicitly relies on similarity transformations to reduce matrices to a block-diagonal structure consisting of Jordan blocks. In his treatise on substitutions and algebraic equations, Jordan demonstrated that every square matrix over the complex numbers is similar to a unique Jordan canonical form, where the blocks correspond to eigenvalues and generalized eigenspaces, enabling the decomposition of linear operators into semisimple and nilpotent parts.^[38] This result, rooted in the study of finite groups and their representations, marked a shift toward using similarity to reveal the intrinsic algebraic structure of matrices, independent of basis choice, and became a cornerstone for solving systems of linear differential equations. In the early 20th century, the spectral theorem extended these ideas to specific classes of matrices under unitary similarity. David Hilbert's work around 1904–1906 established that bounded self-adjoint operators on Hilbert spaces (or equivalently, normal matrices) are unitarily similar to diagonal matrices, with the diagonal entries being the real eigenvalues.^[39] This theorem, developed in the context of integral equations, highlighted how unitary similarity preserves inner products and norms, providing a geometric interpretation of spectral decomposition. Complementing this, Issai Schur's 1909 triangularization theorem proved that every complex square matrix is unitarily similar to an upper triangular matrix, with eigenvalues on the diagonal, offering a more general reduction than full diagonalization and applicable to non-normal matrices. Mid-20th-century progress focused on computational realization of these theoretical reductions, exemplified by Cornelius Lanczos's 1950 algorithm for approximating eigenvalues through iterative similarity transformations. The Lanczos method generates a sequence of tridiagonal matrices via orthogonal similarity reductions of the original symmetric matrix, converging to extremal eigenvalues efficiently for large sparse systems. This approach bridged abstract algebra with practical numerical linear algebra, enabling eigenvalue computations essential for quantum mechanics and engineering applications. In modern extensions, similarity transformations play a central role in the representation theory of Lie groups, particularly the special linear group SL(n, ℂ), where equivalence of representations is defined via conjugation by invertible matrices—precisely similarity. Representations of SL(n, ℂ) preserve volume up to scale (determinant 1), and similarity classifications identify irreducible representations, such as the fundamental representation on ℂⁿ, facilitating applications in physics and geometry.^[40]

References

[1]
Similarity Transformation -- from Wolfram MathWorld
A similarity transformation is a geometric similarity or a matrix transformation that transforms objects into similar objects. It is a conformal mapping.
[2]
Similarity -- from Wolfram MathWorld
A similarity is a transformation that preserves angles and changes all distances in the same ratio, creating similar figures.
[3]
[PDF] Section 47. Similarity Transformations and Results
Jan 5, 2022 · A similarity transformation is a mapping where distances between points change in a fixed ratio. Direct and opposite types exist, and they are ...
[4]
[PDF] SIMILARITY Euclidean Geometry can be described as a study of the ...
Similarity geometry studies properties unchanged by similarity transformations, which preserve shapes but change scales, like magnifying or contracting.
[5]
None
### Summary of Similarity Transformations from Chapter 6
[6]
[PDF] Lecture 23.
Apr 20, 2010 · General properties of similarity transformations. 1. Any isometry is a similarity transformation with ratio 1. 2. Composition S○T of ...Missing: geometry | Show results with:geometry
[7]
Similarity | High school geometry | Math - Khan Academy
Learn what it means for two figures to be similar, and how to determine whether two figures are similar or not. Use this concept to prove geometric theorems.Solve similar triangles (basic) · Determine similar triangles · Solving similar triangles
[8]
Proving Similar Triangles - MathBitsNotebook(Geo)
To prove two triangles are similar, it is sufficient to show that two angles of one triangle are congruent to the two corresponding angles of the other triangle ...
[9]
[PDF] Similarity. Draft
Theorem D. A similarity transformation of a plane is invertible. Proof. By C, any similarity transformation T is a omposition of an isometry.
[10]
[PDF] Similarity
Proposition 171 Every similarity with distinct fixed points is an isometry; a similarity with three distinct non-collinear fixed points is the identity. 113 ...
[11]
[PDF] Fractals, Self-similarity and Hausdorff Dimension
Aug 31, 2016 · In this lecture we construct self-similar sets of fractional dimension. The most basic fractal is the Middle Thirds Cantor Set.
[12]
10.5 Tessellations - Contemporary Mathematics | OpenStax
Mar 22, 2023 · A regular tessellation means that the pattern is made up of congruent regular polygons, same size and shape, including some type of movement; ...
[13]
Introduction to Computer Graphics, Section 2.3 -- Transforms
If you want to scale about a point other than (0,0), you can use a sequence of three transforms, similar to what was done in the case of rotation. A 2D ...
[14]
Dilations - MathBitsNotebook(Geo)
A dilation is a transformation, DO,k , with center O and a scale factor of k that maps O to itself, and any other point P to P'. The center O is a fixed point.Missing: (h, | Show results with:(h,
[15]
Similarity
As suggested by its name, similarity is what is called an equivalence relation. This means that it satisfies the following properties. Proposition. Let A ...
[16]
[PDF] Matrix of a linear transformation. Similar matrices.
Two n×n matrices are similar if and only if they represent the same linear operator on Rn with respect to different bases. Theorem Similarity is an equivalence ...
[17]
Similar Matrices and Their Properties - Ximera
Sep 26, 2024 · The following theorem shows that similarity () satisfies reflexive, symmetric, and transitive properties. Similarity is an equivalence relation.
[18]
[PDF] Linear algebra and the geometry of quadratic equations Similarity ...
Similarity transformations can be thought of in terms of a change of basis (see ... Determine the geometric interpretation of multiplication by a matrix of.
[19]
[PDF] 5.5 Similarity and Diagonalization - Emory Mathematics
Two matrices A and B are similar if B = P−1AP for some invertible P. A is diagonalizable if and only if it is similar to a diagonal matrix.<|control11|><|separator|>
[20]
[PDF] Part IB - Linear Algebra - Dexter Chua
For example, we will show that the rank, trace, determinant and characteristic polynomial are all such invariants. ... Jordan normal form, then the minimal ...<|control11|><|separator|>
[21]
[PDF] MATHEMATICS 217 NOTES - Math (Princeton)
Note that similar matrices have the same characteristic polynomial, since det(λI −C−1AC) = det C−1(λI −A)C = det(λI − A). It is possible to substitute the ...Missing: rank nullity
[22]
[PDF] The QR Algorithm - Ethz
The QR algorithm computes a Schur decomposition of a matrix. It is certainly one of the most important algorithm in eigenvalue computations [9].
[23]
[PDF] 10. Schur decomposition
• similarity transformation preserves eigenvalues and algebraic multiplicities. • if 𝑥 is an eigenvector of 𝐴 then 𝑦 = 𝑋. −1. 𝑥 is an eigenvector of 𝐵 ...
[24]
[PDF] A Numerically Stable Dynamic Mode Decomposition Algorithm for ...
The proposed method comple- ments the DMD for cases where eigendecomposition is ill- conditioned. Both mathematical analysis and the results of numerical ...<|separator|>
[25]
Is there a relationship between SVD and similarity transformation?
Nov 11, 2013 · Is there a relationship between SVD and similarity transformation? I mean, in articles I read about the SVD method, I came across the equation A ...Understanding the singular value decomposition (SVD)Singular Value Theory - linear algebra - Math Stack ExchangeMore results from math.stackexchange.com
[26]
[PDF] Math 4571 (Advanced Linear Algebra)
In this lecture, we discuss how to use diagonalization and the. Jordan canonical form to solve systems of ordinary linear differential equations with constant ...
[27]
[PDF] A Tutorial on Principal Component Analysis
Principal component analysis (PCA) is a mainstay of modern data analysis - a black box that is widely used but poorly understood.<|separator|>
[28]
[PDF] 4 Transformations and Symmetries - DAMTP
As for any group of transformations, in quantum mechanics the group of rotations is represented on H by unitary operators. We denote the unitary operator ...
[29]
Proportionality in Similar Triangles: A Cross-Cultural Comparison
Similarity has its roots in antiquity, and ideas of proportion were likely known to the Pythagoreans in the fifth century [4, p. 82-83], while Eudoxus of Cnidus ...Missing: analysis | Show results with:analysis
[30]
Euclid's Elements, Book VI - Clay Mathematics Institute
May 8, 2008 · Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon ...Missing: analysis | Show results with:analysis
[31]
[PDF] The Mathematics of Perspective Drawing: From Vanishing Points to ...
Filippo Brunelleschi (1377 – 1446) was one of the first to discover the rules of perspective. He used a mirror to demonstrate the accuracy of his paintings ...
[32]
[PDF] Projective Geometry: A Short Introduction - Morpheo
19th century : Poncelet (a Napoleon officer) writes, in 1822, a treaty on projective properties of figures and the invariance by projection. This is the ...
[33]
The Values of Simplicity and Generality in Chasles's ... - jstor
May 29, 2019 · 41 Homotheties (a term introduced by Chasles himself) refer to figures which derive from one another by a homogeneous dilation. Here, the second ...
[34]
[PDF] Felix Klein and his Erlanger Programm
The choice of distinct transformation groups leads to distinct geometries. Thus, the analysis of the group of motions leads to the common. Euclidean geometry.
[35]
II. A memoir on the theory of matrices - Journals
It will be seen that matrices (attending only to those of the same order) comport themselves as single quantities; they may be added, multiplied or compounded ...
[36]
[PDF] A Brief History of Linear Algebra and Matrix Theory
Cayley studied compositions of linear transformations and was led to define matrix multiplication so that the matrix of coefficients for the composite ...Missing: similarity | Show results with:similarity
[37]
[PDF] Cayley, Sylvester, and Early Matrix Theory
Matrix algebra developed by. Arthur Cayley, FRS (1821–. 1895). Memoir on the Theory of Ma- trices (1858). MIMS. Nick Higham.<|separator|>
[38]
A history of the Jordan decomposition theorem (1870-1930) Forms ...
The thesis takes as its point of departure the Jordan decomposition theorem and traces its evolution over the sixty-year period from its statement by Camille ...
[39]
Highlights in the History of Spectral Theory - jstor
Hilbert's original spectral theorem applied to real quadratic forms (or infinite matrices) that were bounded and symmetric. This theorem was quickly and easily.Missing: similarity | Show results with:similarity
[40]
[PDF] Representations of Semisimple Lie Groups
The context for this section is an overview of one of the fundamental problems in the representation theory of Lie groups: the problem of determining the “ ...