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Basic Definitions

The matrix basics can be found on the linear algebra page.

Linear Algebra

Permutation

In general a permutation is a rearrangement of elements of an ordered list like the elements of a vector. Permutations can be expressed as matrix operations on a vector. If an arrangement contains n elements n! different permutations are possible.

Permutaions as Matrix Operations

A permutation is a matrix operation that leads two the exchange of the elements of a vector, when the matrix is multiplied with the vector.

The unit matrix is a permutation matrix, that leaves the order of the vector elements unchanged when applied to the vector.

Exchange Matrix

The exchange Matrix is the next example of a permutation matrix.

The exchange matrix or counterbrace matrix J_n, is a matrix where the counterdiagonal elements are all one. All other elements are zero.

The product of the exchage matrix with a column vector will result in a reordering of the column elements. The product of a row vector with a exchange matrix will result in a reordering of the row elements.

Square of the Exchange Matrix

J² = I

The square of the exchange matrix is the unit matrix. Applying the exchange matrix twice will result in the original matrix.

Transpositions

Transpositions are permutations where only two rows of a matrix are exchanged. Every permutation can be expressed as a product of transposions.

Transpositions with a 2*2 matrix:

With two elements 2! = 2 permutations are possible.

Transpositions with a 3*3 matrix

With three elements 3! = 6 permutations are possible.

Factorisation of a Matrix A in Triangular Matrices A=LU

U is a upper triangular matrix and L is a lower triangular matrix:

Elimination Matrix

With the elemintion matrix E₂₁ it is possible to bring a 2×2 matrix A into the upper triangular form U.

Example

A=(E₂₁)^-1U = LU

(E₂₁)^-1 is the lower triangular matrix L.

Nilpotent Matrices

Matrix A is called nilpotent of grade n when Aⁿ=0.

Example 3x3 Matrices

Example 4x4 Matrices

The next page is about rotation matrices.

Rotation Matrices