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• 1.2 Matrix algebra

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• How are matrices formulated?

  • Rows are always named first, and columns are named second
    
  • also in naming specific cells with subscript; X34 is the cell in row three column 4.

• What is the transpose of a matrix?

  • The transpose of a matrix A is the same matrix but the dimensions are inverted; a 3x4 matrix becomes a 4x3 matrix
  • it is done by making thee first row, the first column. The second row becomes the second column, etc.
  • the transpose of matrix A is indicated by A'

• When is a matrix symmetric?

  • A matrix is symmetric if:
    
  • the amount of columns is equal to the amount of rows
  • the elements below the main diagonal of the matrix are mirroring the elements above the main diagonal

  
  
  

  • a12 = a21, a31 = a13, etc
  • the matrix can be folded in half along the diagonal and the two parts would be fitting onto each other perfectly.

  
  

  • a frequently encountered symmetric matrix is a correlation matrix, since the correlation between var 1 andd var 2 is equal to the correlation between var 2 and var 1

• How can two matrices be added or subtracted?

  • Two matrices can be added or subtracted by adding or subtracting each element of the first matrix to the element in the same position of the second matrix.
    
  • therefor only matrices that have identical dimensions can be added or subtracted to each other.

• How can a matrix be multiplied by a scalar (number)?

  Each element of the matrix is multiplied by the scalar.
  the positions of the elements are unchanged.

• What is the order in which we can multiply more than two matrices?

  • Multiplying more than two matrices results in the same result for different orders;
    
  • A * B * C can be done by (A * B) * C, or A * (B * C)

• What is the determinant of a matrix?

  • The determinant of a matrix A is denoted by |A|.
  • the determinant of a covariance matrix represents the generalized variance of the conjunction of those variables;
  • it characterizes in a single number how much variance is present in a set of variables.
  • the determinant, notation of generalized variance, is used in a lot of multivariate statistical tests

• How is the determinant of a variance-covariance matrix influenced by the presence or absence of covariance between the variables?

  • The determinant denotes general variance in the matrix.
    
  • if there is covariance present in the covariance matrix, some of the variance in variable 1 is accounted for by variance in variable 2. This causes the determinant to become lower.
  • if there is no covariance between variables, no variance in either variable can be accounted for by the other variable, which causes the generalized variance to increase.

• What is the formula for variance of a variable?

  • (Sum(xi - mean(x))) / (n-1)
    
  • the variance is thus the sum of deviances from the mean for each component of the variable, divided by degrees of freedom of the variable.

• How can you obtain a statistic that describes the general variance of a group of participants for a set of variables?

  • Taking the determinant of the variance-covariance matrix describes how much generalized variance is in the sample for those variables.
    
  • for one variable, variance can be described as the spread of points across a line, for two variables the spread over a plane, and for three variables the spread within a volume.