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2. Arnold's cat map - Wikipedia

   en.wikipedia.org/wiki/Arnold's_cat_map

   The linear transformation which defines the map is hyperbolic: its eigenvalues are irrational numbers, one greater and the other smaller than 1 (in absolute value), so they are associated respectively to an expanding and a contracting eigenspace which are also the stable and unstable manifolds.

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. 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,

5. Vectorization (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Vectorization_(mathematics)

   The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, vec ⁡ ( A B C ) = ( C T ⊗ A ) vec ⁡ ( B ) {\displaystyle \operatorname {vec} (ABC)=(C^{\mathrm {T} }\otimes A)\operatorname {vec} (B)} for matrices A , B , and C of dimensions k ...

6. File:Linear transformation.pdf - Wikipedia

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

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7. Square matrix - Wikipedia

   en.wikipedia.org/wiki/Square_matrix

   In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order . Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as shearing or rotation.

8. Linear least squares - Wikipedia

   en.wikipedia.org/wiki/Linear_least_squares

   Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approximate solution is realized as an exact solution to A x = b', where b' is the projection of b onto the column space of A. The best ...

9. 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 ...