{"id":16698,"date":"2020-01-29T23:10:58","date_gmt":"2020-01-30T04:10:58","guid":{"rendered":"http:\/\/bangla.salearningschool.com\/recent-posts\/?p=16698"},"modified":"2020-02-08T09:40:24","modified_gmt":"2020-02-08T14:40:24","slug":"misc-math-data-science-machine-learning-pca-fa","status":"publish","type":"post","link":"http:\/\/bangla.sitestree.com\/?p=16698","title":{"rendered":"Misc Math, Data Science, Machine Learning, PCA, FA"},"content":{"rendered":"

“In mathematics<\/a>, a set B of elements (vectors) in a vector space<\/a> V<\/em> is called a basis<\/strong>, if every element of V<\/em> may be written in a unique way as a (finite) linear combination<\/a> of elements of B. The coefficients of this linear combination are referred to as components<\/strong> or coordinates<\/strong> on B of the vector. The elements of a basis are called basis vectors<\/strong>.”<\/p>\n

”<\/p>\n

Equivalently B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B.[1]<\/a> In more general terms, a basis is a linearly independent spanning set<\/a>.<\/p>\n

A vector space can have several bases; however all the bases have the same number of elements, called the dimension<\/a> of the vector space.<\/p>\n

”<\/p>\n

https:\/\/en.wikipedia.org\/wiki\/Basis_(linear_algebra)<\/a><\/p>\n

Positive Semidefinite Matrix<\/strong>
\n“A positive semidefinite<\/strong> matrix is a Hermitian matrix all of whose eigenvalues are nonnegative. SEE ALSO: Negative Definite Matrix, Negative Semidefinite<\/strong> Matrix, Positive<\/strong> Definite Matrix, Positive<\/strong> Eigenvalued Matrix, Positive<\/strong> Matrix.”<\/p>\n

http:\/\/mathworld.wolfram.com\/PositiveSemidefiniteMatrix.html<\/a><\/p>\n

”<\/p>\n

Hermitian Matrix<\/h1>\n

A square matrix<\/a> is called Hermitian if it is self-adjoint<\/a>. Therefore, a Hermitian matrix \"A=(a_(ij))\" is defined as one for which<\/p>\n\n\n\n
\"A=A^(H),\"<\/td>\n(1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where \"A^(H)\" denotes the conjugate transpose<\/a>. This is equivalent to the condition<\/p>\n\n\n\n
\"a_(ij)=a^__(ji),\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

”<\/p>\n

http:\/\/mathworld.wolfram.com\/HermitianMatrix.html<\/a><\/p>\n

Definiteness of a matrix<\/h1>\n

“In linear algebra<\/a>, a symmetric<\/a> \"n\\times real<\/a> matrix<\/a> \"M\" is said to be positive definite<\/strong> if the scalar \"{\\displaystyle is strictly positive for every non-zero column vector<\/a> \"z\" of \"n\" real numbers. Here \"{\\displaystyle denotes the transpose<\/a> of \"z\".[1]<\/a> When interpreting \"{\\displaystyle as the output of an operator, \"M\", that is acting on an input, \"z\", the property of positive definiteness implies that the output always has a positive inner product<\/a> with the input, as often observed in physical processes.”
\nhttps:\/\/en.wikipedia.org\/wiki\/Definiteness_of_a_matrix<\/a><\/p>\n

”<\/p>\n

Singular value decomposition<\/h1>\n

From Wikipedia, the free encyclopedia<\/p>\n

Jump to navigation<\/a>Jump to search<\/a>\"\"<\/a>
\nIllustration of the singular value decomposition U\u03a3V<\/strong>* of a real 2\u00d72 matrix M<\/strong>.<\/p>\n