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Matrix Algebra and Regression

Matrix Algebra and Regression. a matrix is a rectangular array of elements m=#rows, n=#columns  m x n a single value is called a ‘scalar’ a single row is called a ‘row vector’ a single column is called a ‘column vector’. matrix element. a 13 = 6. B = 12 25 91 30.

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Matrix Algebra and Regression

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  1. Matrix Algebra and Regression • a matrix is a rectangular array of elements • m=#rows, n=#columns  m x n • a single value is called a ‘scalar’ • a single row is called a ‘row vector’ • a single column is called a ‘column vector’ matrix element a13 = 6 B = 12 25 91 30

  2. Matrix Algebra and Regression • a square matrix has equal numbers of rows and columns • in a symmetric matrix, aij = aji • in a diagonal matrix, all off-diagonal elements = 0 • an identity matrix is a diagonal matrix with diagonals = 1 I =

  3. Trace • The trace of a matrix is the sum of the elements on the main diagonal A = tr(A) = 2 + 6 + 3 + 1 + 8 = 20

  4. Matrix Addition and Subtraction • The dimensions of the matrices must be the same

  5. Matrix Multiplication C11 = 2*2 + 5*5 + 1*1 + 8*8 = 94 A m x n C m x p B n x p = X • The number of columns in A must equal the number of rows in B • The resulting matrix C has the number of rows in A and the number of columns in B • Note that the commutative rule of multiplication does not apply to matrices: A x B ≠ B x A

  6. Transpose a Matrix • Multiplying A x A′ above will give the uncorrected sums of squares for each row in A on the diagonal of a 2 x 2 matrix, with the sums of crossproducts on the off-diagonals

  7. Invert a Matrix • The inverse of a matrix is analogous to division in math • An inverted matrix multiplied by the original matrix will give the identity matrix M-1M = M-1M =I • It is easy to invert a diagonal matrix:

  8. Inverting a 2x2 Matrix • Calculate the Determinant (D) of the matrix M • Verify • The extension to larger matrices is not simple – use a computer! M = M = |M| = D = ad - bc D = 2*9 – 5*3 M-1 = M-1 =

  9. Linear Dependence M = M = D = ad - bc D = 2*9 – 6*3 = 0 The matrix M on the right is singular because one row (or column) can be obtained by multiplying another by a constant. A singular matrix will have D=0. The rank of a matrix = the number of linearly independent rows or columns (1 in this case). A nonsingular matrix is full rank and has a unique inverse. A generalized inverse (M–) can be obtained for any matrix, but the solution will not be unique if the matrix is singular. MM–M = M

  10. Regression in Matrix Notation Linear model Y = X + ε Parameter estimates b = (X’X)-1X’Y

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