Matrices

• 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 M_ij 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.

Transponse

• • The transpose of M is known as M^T. It is the c*r matrix where the columns are formed from the rows of M. It 'flips' the matrix diagonally. M^T_ij = M_ij

    • ₍M^T₎^T = M. The transpose of a matrix transpose is the matrix again.

  • D^T = D for any diagonal matrix.

Matrix Multiplying

• 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 C_ij 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

Determinant

• • 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|

  • |M^T| = |M|

Inverse

• • 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

• (M^T)^-1 = (M^-1)^T

Orthogonal

• • A square matrix is orthogonal if the product of the matrix and it's transpose is the identity matrix. MM^T = I

  • If a matrix is orthogonal then the inverse and the transpose is equal. M^T = M^-1

Reference

• • Naming Convention for Matrix Math - 2017

  • Matrix Multiplication - 2016

  • Matrix maths and names - 2012

  • 3D Math Primer for Game Programmers (Matrices) - 2011

  • Complex Matrix Transformations - 2002

  • The Matrix and Quaternions FAQ - 2000

• Matrices can be your Friends.