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Introduction to Matrix Algebra:

In mathematics, a Matrix (plural is matrices) is a rectangular array of symbols, expressions or numbers, which are arranged in rows and columns. Separate items in a matrix are known as its elements or entries. An illustration of a matrix with 2 rows and 3 columns is as follows:

The Matrices of similar size can be added or subtracted element by element. The law for matrix multiplication is more complex and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. A most significant application of matrices is to represent the linear transformations, that is, the generalizations of linear functions like f(x) = 4x. For example, the rotation of vectors in three dimensional spaces is a linear transformation. When M is a rotation matrix and c is a column vector (a matrix with only one column) explaining the position of a point in space, the product Mc is a column vector explaining the location of that point after a rotation, product of two matrices is a matrix which represents the composition of two linear transformations. The other application of matrices is in the solution of a system of linear equations. When the matrix is square than it is possible to deduce few of its properties by calculating its determinant. For illustration, a square matrix has an inverse if and only if its determinant is not 0. The Eigen values and eigenvectors give insight into the geometry of linear transformations.

The Matrices find applications in many scientific fields. In every field of physics, involving electromagnetism, classical mechanics, quantum electrodynamics and quantum mechanics, they are used to study the physical phenomena, such as the motion of firm bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In statistics and probability theory, the stochastic matrices are used to explain sets of probabilities; for illustration, they are used within the Page Rank algorithm and that ranks the pages in a Google search.

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• Hessenberg Form
• Householder Transform
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