This page is the red pill
The size of a matrix is the number of rows and columns.
A matrix with M rows and N columns is a MxN matrix.
Individual elements in the matrix is referenced using two index values. The order is row first then column.
Common matrices used are 2x2, 3x3 and 4x4.
The major diagonal of a matrix is the elements where the row number is equal to the column number. That is Mij where i=j.
If all non diagonal elements in a matrix is zero then the matrix is a diagonal matrix.
The identity matrix I is a matrix that have a size of NxN and all elements i=j is set to one. All others are zero.
The transpose of M is known as MT. It is the c*r matrix where the columns are formed from the rows of M. It 'flips' the matrix diagonally. MTij = Mij
(MT)T = M. The transpose of a matrix transpose is the matrix again.
DT = D for any diagonal matrix.
Multiplying a matrix M with a scalar k ( kM ) is done by multiplying each element in the matrix with the scalar.
Multiplying two matrices
A r*n matrix A may be multiplied by a n*c matrix B. The result C is a r*c matrix. The number of columns in A needs to be the same as the number of columns in B.
Each element Cij will be the sum of the dot product of row i in A with column j in B.
Multiplying any matrix M with a square matrix S result in a matrix of the same size as M. If S is the identity matrix I then the result will be M.
AB != BA
Multiplying a vector and a matrix
The determinant of a square matrix M is a scalar denoted |M|.
It only exist for square matrices.
If any row or column in a matrix is all zero then the determinant of the matrix is zero.
||AB|| = |A| |B|
|MT| = |M|
The inverse of a square matrix M, M-1 is the matrix that when multiplied by M will result in I.
Not all matrices have an inverse.
A matrix that is invertible have a non zero determinant.
MM-1 = I
(M-1)-1 = M
I-1 = I
(MT)-1 = (M-1)T
A square matrix is orthogonal if the product of the matrix and it's transpose is the identity matrix. MMT = I
If a matrix is orthogonal then the inverse and the transpose is equal. MT = M-1