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Matrix Algebra . Matrix algebra is a means of expressing large numbers of calculations made upon ordered sets of numbers. Often referred to as Linear Algebra. Why use it?. Matrix algebra is used primarily to facilitate mathematical expression.

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matrix algebra
Matrix Algebra
  • Matrix algebra is a means of expressing large numbers of calculations made upon ordered sets of numbers.
  • Often referred to as Linear Algebra
why use it
Why use it?
  • Matrix algebra is used primarily to facilitate mathematical expression.
  • Many equations would be completely intractable if scalar mathematics had to be used. It is also important to note that the scalar algebra is under there somewhere.
definitions scalar
Definitions - scalar
  • scalar - a number
    • denoted with regular type as is scalar algebra
    • [1] or [a]
definitions vector
Definitions - vector
  • vector - a single row or column of numbers
    • denoted with bold small letters
    • row vector
    • column vector
definitions matrix
Definitions - Matrix
  • A matrix is a set of rows and columns of numbers
  • Denoted with a bold Capital letter
  • All matrices (and vectors) have an order - that is the number of rows x the number of columns. Thus A is
matrix equality
Matrix Equality
  • Thus two matrices are equal iff (if and only if) all of their elements are identical
  • Note: your data set is a matrix.
matrix operations
Matrix Operations
  • Addition and Subtraction
  • Multiplication
  • Transposition
  • Inversion
addition and subtraction
Addition and Subtraction
  • Two matrices may be added iff they are the same order.
  • Simply add the corresponding elements
matrix multiplication
Matrix Multiplication
  • To multiply a scalar times a matrix, simply multiply each element of the matrix by the scalar quantity
matrix multiplication cont
Matrix Multiplication (cont.)
  • To multiply a matrix times a matrix, we write
      • A times B as AB
  • This is pre-multiplying B by A, or post-multiplying A by B.
matrix multiplication cont12
Matrix Multiplication (cont.)
  • In order to multiply matrices, they must be conformable (the number of columns in A must equal the number of rows in B.)
  • an (mxn) x (nxp) = (mxp)
  • an (mxn) x (pxn) = cannot be done
  • a (1xn) x (nx1) = a scalar (1x1)
matrix multiplication is not commutative
Matrix multiplication is not Commutative
  • AB does not necessarily equal BA
  • (BA may even be an impossible operation)
special matrices
Special matrices
  • There are a number of special matrices
    • Diagonal
    • Null
    • Identity
diagonal matrices
Diagonal Matrices
  • A diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero.
identity matrix
Identity Matrix
  • An identity matrix is a diagonal matrix where the diagonal elements all equal one. It is used in a fashion analogous to multiplying through by "1" in scalar math.
null matrix
Null Matrix
  • A square matrix where all elements equal zero.
  • Not usually ‘used’ so much as sometimes the result of a calculation.
    • Analogous to “a+b=0”
the transpose of a matrix a
The Transpose of a Matrix A'
  • Taking the transpose is an operation that creates a new matrix based on an existing one.
  • The rows of A = the columns of A'
  • Hold upper left and lower right corners and rotate 180 degrees.
the transpose of a matrix a23
The Transpose of a Matrix A'
  • If A = A', then A is symmetric (i.e. correlation matrix)
  • If AA’ = A then A' is idempotent (and A' = A)
  • The transpose of a sum = sum of transposes
  • The transpose of a product = the product of the transposes in reverse order
an example
An example:
  • Suppose that you wish to obtain the sum of squared errors from the vector e. Simply pre-multiply e by its transpose e'.
  • which, in matrices looks like
an example cont
An example - cont
  • Since the matrix product is a scalar found by summing the elements of the vector squared.
the determinant of a matrix
The Determinant of a Matrix
  • The determinant of a matrix A is denoted by |A|.
  • Determinants exist only for square matrices.
  • They are a matrix characteristic, and they are also difficult to compute
the determinant for a 2x2 matrix
The Determinant for a 2x2 matrix
  • If A =
  • Then
  • That one is easy
determinants
Determinants
  • For 4 x 4 and up don't try. For those interested, expansion by minors and cofactors is the preferred method.
  • (However the spaghetti method works well! Simply duplicate all but the last column of the matrix next to the original and sum the products of the diagonals along the following pattern.)
properties of determinates
Properties of Determinates
  • Determinants have several mathematical properties which are useful in matrix manipulations.
    • 1 |A|=|A'|.
    • 2. If a row of A = 0, then |A|= 0.
    • 3. If every value in a row is multiplied by k, then |A| = k|A|.
    • 4. If two rows (or columns) are interchanged the sign, but not value, of |A| changes.
    • 5. If two rows are identical, |A| = 0.
properties of determinates32
Properties of Determinates
  • 6. |A| remains unchanged if each element of a row or each element multiplied by a constant, is added to any other row.
  • 7. Det of product = product of Det's |A| = |A| |B|
  • 8. Det of a diagonal matrix = product of the diagonal elements
the inverse of a matrix a 1
The Inverse of a Matrix (A-1)
  • For an nxn matrix A, there may be a B such that AB = I = BA.
  • The inverse is analogous to a reciprocal)
  • A matrix which has an inverse is nonsingular.
  • A matrix which does not have an inverse is singular.
  • An inverse exists only if
inverse by row or column operations
Inverse by Row or column operations
  • Set up a tableau matrix
  • A tableau for inversions consists of the matrix to be inverted post multiplied by a conformable identity matrix.
matrix inversion by tableau method
Matrix Inversion by Tableau Method
  • Rules:
    • You may interchange rows.
    • You may multiply a row by a scalar.
    • You may replace a row with the sum of that row and another row multiplied by a scalar.
  • Every operation performed on A must be performed on I
  • When you are done; A = I& I = A-1
matrix inversion cont
Matrix Inversion – cont.
  • Step 2: Add –2(Row 1) to Row 2
  • Step 3: Add –1(Row 1) to Row 3
matrix inversion cont38
Matrix Inversion – cont.
  • Step 4: Multiply Row 2 by –1/3
  • Step 5: Add –4 (Row 2) to Row 1
matrix inversion cont39
Matrix Inversion – cont.
  • Step 6: Add 7(Row 2) to Row 3
  • Step 7: Add Row 3 to Row 1
matrix inversion cont40
Matrix Inversion – cont.
  • Step 9: Add 2(Row 3) to Row 2
  • Step 9: Multiply Row 3 by -3
checking the calculation
Checking the calculation
  • Remember AA-1=I
  • Thus
the matrix model
The Matrix Model
  • The multiple regression model may be easily represented in matrix terms.
  • Where the Y, X, B and e are all matrices of data, coefficients, or residuals
the matrix model cont
The Matrix Model (cont.)
  • The matrices in are represented by
  • Note that we postmultiply X by B since this order makes them conformable.
the assumptions of the model scalar version
The Assumptions of the ModelScalar Version
  • 1. The ei's are normally distributed.
  • 2. E(ei) = 0
  • 3. E(ei2) = 2
  • 4. E(eiej) = 0 (ij)
  • 5. X's are nonstochastic with values fixed in repeated samples and (Xik-Xbark)2/n is a finite nonzero number.
  • 6. The number of observations is greater than the number of coefficients estimated.
  • 7. No exact linear relationship exists between any of the explanatory variables.
the assumptions of the model the matrix version
The Assumptions of the Model: The Matrix Version
  • These same assumptions expressed in matrix format are:
    • 1. e  N(0,)
    • 2.  = 2I
    • 3. The elements of X are fixed in repeated samples and (1/ n)X'X is nonsingular and its elements are finite