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2. Proofs involving ordinary least squares - Wikipedia

   en.wikipedia.org/wiki/Proofs_involving_ordinary...

   Recall that M = I − P where P is the projection onto linear space spanned by columns of matrix X. By properties of a projection matrix, it has p = rank(X) eigenvalues equal to 1, and all other eigenvalues are equal to 0. Trace of a matrix is equal to the sum of its characteristic values, thus tr(P) = p, and tr(M) = n − p. Therefore,

3. Transformation matrix - Wikipedia

   en.wikipedia.org/wiki/Transformation_matrix

   In linear algebra, linear transformations can be represented by matrices.If is a linear transformation mapping to and is a column vector with entries, then there exists an matrix , called the transformation matrix of , [1] such that: = Note that has rows and columns, whereas the transformation is from to .

4. File:Linear transformation.pdf - Wikipedia

   en.wikipedia.org/wiki/File:Linear_transformation.pdf

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5. Trace (linear algebra) - Wikipedia

   en.wikipedia.org/wiki/Trace_(linear_algebra)

   In general, given some linear map f : V → V (where V is a finite-dimensional vector space), we can define the trace of this map by considering the trace of a matrix representation of f, that is, choosing a basis for V and describing f as a matrix relative to this basis, and taking the trace of this square matrix.

6. Matrix similarity - Wikipedia

   en.wikipedia.org/wiki/Matrix_similarity

   A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however, in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than ...

7. Conformal linear transformation - Wikipedia

   en.wikipedia.org/.../Conformal_linear_transformation

   Conformal linear transformations come in two types, proper transformations preserve the orientation of the space whereas improper transformations reverse it. As linear transformations, conformal linear transformations are representable by matrices once the vector space has been given a basis , composing with each-other and transforming vectors ...

8. Linear map - Wikipedia

   en.wikipedia.org/wiki/Linear_map

   A specific application of linear maps is for geometric transformations, such as those performed in computer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to ...

9. Matrix equivalence - Wikipedia

   en.wikipedia.org/wiki/Matrix_equivalence

   In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if = for some invertible n-by-n matrix P and some invertible m-by-m matrix Q.Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.