Two shapes are called similar if each corresponding angle is congruent and all distances are either increased or decreased by the same ratio. This ratio is known as the ratio of magnification. Any transformation that takes a shape to a similar shape is called a similarity, or a similarity transformation. Two shapes are directly similar if the angles correspond in the same rotational order, or inversely similar if they correspond in the opposite rotational order. For example:

|\  is directly similar to  |\
|/                          | \
                            | /
                            |/

|\  is inversely similar to  /|
|/                          / | 
                            \ | 
                             \|

Two matrices A and B are similar if there is an invertible matrix P such that P-1AP = B. One way to look at similar matrices is that they represent the same linear map with different bases. Similar matrices have the same rank, trace, determinant, eigenvalues, characteristic polynomial and minimal polynomial. If the eigenvalues of A are not degenerate, then B can be diagonal. If not, then B can be in Jordan form (if not with all real elements, then with complex elements). This is useful for finding solutions to systems of linear equations. In this case, P is a matrix of eigenvectors (or generalized eigenvectors), and A is the coefficient matrix.

Sources:
Eric W. Weisstein. "Similar." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/Similar.html
http://en.wikipedia.org/wiki/Similar

Sim"i*lar (?), a. [F. similaire, fr. L. similis like, similar. See Same, a., and cf. Simulate.]

1.

Exactly corresponding; resembling in all respects; precisely like.

2.

Nearly corresponding; resembling in many respects; somewhat like; having a general likeness.

3.

Homogenous; uniform.

[R.]

Boyle.

Similar figures Geom., figures which differ from each other only in magnitude, being made up of the same number of like parts similarly situated. -- Similar rectilineal figures, such as have their several angles respectively equal, each to each, and their sides about the equal angles proportional. -- Similar solids, such as are contained by the same number of similar planes, similarly situated, and having like inclination to one another.

 

© Webster 1913.


Sim"i*lar, n.

That which is similar to, or resembles, something else, as in quality, form, etc.

 

© Webster 1913.

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