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
**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*(*k***M**) is done by multiplying each element in the matrix with the scalar.
- 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**
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 |

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