Matrices

Difference Matrix

A combination can ne written as matrix of form Ax = b. It implies that the matrix A acts on the vector x. The result Ax is a combination b of the columns of A. The input is x and the output is b = Ax. This A is a "difference matrix" because b contains differences of the input vector x.

Invertible Matrix

An Invertible Matrix is a square matrix defined as invertible if the product of the matrix and its inverse is the identity matrix.

Inverse of a Matrix

Inverse of a matrix is a matrix that when multiple by a matrix gives an identity matrix.

Differences and Sums in Ax = b

If the differences of the x's are the b's, the sums of the b's are the x's. That was true for the odd numbers b = (1,3,5) and the squares x = (1,4,9). It is true for all vectors. The sum matrix S is the inverse of the difference matrix A.

Ax = b and x = Sb

Note on calculus

Let me connect these special matrices A and S to calculus. The vector x changes to a function x(t). The differences Ax become the derivative dx/dt = b(t). In the inverse direction, the sum Sb becomes the integral of b(t). The Fundamental Theorem of Calculus says that integration S is the inverse of differentiation A.

Cyclic Difference

Cyclic Difference Matrix

Cyclic difference matrix is of the following form

It is impossible to find the solution to Cx = b, because the three equations either have infinitely many solutions or else no solution. Where C is the cyclic matrix.

Dependence and Independence

The vectors go into the columns of an n by n matrix:

• Independent columns: Ax = 0 has one solution. A is an invertible matrix.
• Dependent columns: Ax = 0 has many solutions. A is a singular matrix.

More pointers:

• Each of the vectors has a different plane means there is just one way to get b = 0 (Ax = b), by using 0 as coefficients, i.e., x = (0, 0, 0). Vectors are independent.

• Vectors have shared planes means there are more ways to get b = 0 (Ax = b), then just using 0 as coefficients, i.e. x = (0, 0, 0). Vectors are dependent.

Inverse of a matrix

(See more details here)[https://www.mathsisfun.com/algebra/matrix-inverse.html]

Transpose Matrix

Replacing rows with columns.

Permutation Matrix

A permutation matrix P has the rows of the identity matrix in any order. It is used to permute another matrix.

Determinant

Details here