Download APPENDIX Matrix Algebra

APPENDIX Econometrics I – Matrix Algebra
Ing. Martina Hanová, PhD.
MATRIX ALGEBRA REVIEW
MATRIX DEFINITIONS
A MATRIX is a rectangular collection of numbers. An r × c matrix (also called an array) is a
rectangular array (or table) with r rows and c columns.
1
2
5 4
5 3
Matrix A is a matrix with two rows and three columns. For that reason, it is called a 2 by 3
matrix. This is called the dimension of a matrix. It is expressed as number of rows × number
of columns. So, B is a 4 × 3 matrix.
1
5
1
2
5
3
5
5
4
8
4
3
It is common to refer to elements in a matrix by subscripts. We can represent such a matrix
as
B = [bij]
where
1≤i≤r
and
1 ≤ j ≤ c.
Thus bij is the element in the ith row and jth column.
with the row first and the column second. So in this case, b2,1= 5 and b1,3= 4. Generally,
matrices are denoted as bold capital letters.
A SQUARE MATRIX is one where r = c. The main diagonal of a square k × k matrix A is a11, …,
akk. If all the elements in the square matrix A are 0 then the matrix is called the null matrix.
An identity matrix is a square matrix where on the main diagonal are 1 and all other
elements are 0, denoted I (or Ik to emphasize that it is a k × k identity matrix). The following
two matrices are both identity matrices and diagonal matrices.
1
0
0
0 0
1 0
0 1
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APPENDIX Econometrics I – Matrix Algebra
Ing. Martina Hanová,
Hanov PhD.
They are called identity matrices, because any matrix multiplied with an identify matrix
equals itself. The diagonal entries of a matrix are the entries where the column and row
number are the same.
The trace of a n × n matrix is the sum of all the diagonal entries. In other words, for n × n
matrix A, The trace of A is a11 + … + akk.
A triangular matrix has either all zeros in the triangular portion below the main diagonal (an
upper triangular matrix) or all zeros in the triangular portion above the main diagonal (a
lower triangular matrix).
Define matrix A = [aij] as follows:
A is a 3 × 3 square matrix whose main diagonal is 3, 5, 2. The trace of is 3 + 5 + 2 = 10. The
element in the 2nd row and 3rd column is a23 = 4. A is an upper triangular matrix.
Two matrices A = [aij] and B = [bij] are equal, denoted A = B, if they have the same size and
shape (i.e. the same number of
of rows and columns) and all the corresponding elements are
equal (i.e. aij = bij for all i, j with 1 ≤ i ≤ r and 1 ≤ j ≤ c).
EXCEL: a matrix is represented as a rectangular range. E.g. B3:E5
B3: can be considered to be a 3
× 4 matrix.
A VECTOR is a matrix with only one row (called a row vector) or only one column (called a
column vector). For example y is a 4 dimensional row vector and x is a 3 dimensional column
vector.
An "ordinary" number can be thought of as a 1 × 1 matrix, also known as a SCALAR.
E =π or
K =6
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APPENDIX Econometrics I – Matrix Algebra
Ing. Martina Hanová, PhD.
MATRIX ARITHMETIC
To take the TRANSPOSE of a matrix, simply switch the rows and column of a matrix. The
transpose of A can be denoted as A′ or AT. For example, for
1
2
5
5
4
3
1 2
5 5
4 3
EXCEL: If A is an 2 × 3 array, then TRANSPOSE(A) = AT. This is an array function and so you
must highlight an 3 × 2 range before entering =TRANSPOSE(A) and then pressing Ctrl-ShiftEnter. If you click on OK or enter, press F2 and then Ctrl-Shift-Enter
You can also transpose an array A in Excel by copying the array (i.e. by highlighting the array
of A and pressing Ctrl-C), clicking where you want AT located (i.e. the cell at the upper left
corner of AT) and then selecting Home/Paste/ and choosing the Transpose option.
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APPENDIX Econometrics I – Matrix Algebra
Ing. Martina Hanová, PhD.
If a matrix is its own transpose, then that matrix is said to be symmetric. Symmetric matrices
must be square matrices, with the same number of rows and columns. One example of a
symmetric matrix is shown below:
1
5
4
5 4
7 3
3 3
To perform MATRIX ADDITION, two matrices must have the same dimensions. This means
they must have the same number of rows and columns. In that case simply add each
individual component, like below.
EXCEL: Perform matrix addition in Excel create formula by the address of the cells for first
element of A+B and use copy function to copy formula for rest of the matrix elements.
Matrix addition does have many of the same properties as "normal" addition, such as
In addition, if one wishes to take the transpose of the sum of two matrices, then
To multiply a matrix by a scalar, also known as SCALAR MULTIPLICATION, multiply every
element in the matrix by the scalar. For example:
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APPENDIX Econometrics I – Matrix Algebra
Ing. Martina Hanová, PhD.
EXCEL: Perform scalar multiplication in Excel create formula for first element of [k*A] matrix
and use copy function to copy formula for rest of the matrix elements.
To multiply two vectors with the same length together is to take the SUM PRODUCT, also
called dot or inner product. This is done by multiplying every entry in the two vectors
together and then adding all the products up. For example, for vectors x and y, the dot
product is calculated below
EXCEL: Multiply two vectors in Excel use function SUMPRODUCT(x,y). In this context an array
is a continuous range of cells in a row or column. The number of cells must be the same in
each array.
To perform MATRIX MULTIPLICATION, the first matrix must have the same number of
columns as the second matrix has rows. The number of rows of the resulting matrix equals
the number of rows of the first matrix, and the number of columns of the resulting matrix
equals the number of columns of the second matrix. So a 2 × 3 matrix could be multiplied by
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APPENDIX Econometrics I – Matrix Algebra
Ing. Martina Hanová, PhD.
a 3 × 4 matrix, forming a 2 × 4 matrix, but one cannot multiply a 3 × 4 matrix with a 2 × 3
matrix.
3
2
9
4
8
3
7
2
6
3
3
2
2 6
4 2
3 5
9
24
52 6
12 11
40
35
EXCEL: Multiply two matrix in Excel use function MMULT(A, B). If A is an p × m array and B is
an m × n array, then MMULT(A, B) = the p × n matrix A*B. Note that since this is an array
function, you must first highlight a p × n range before entering =MMULT(A, B) and then you
must press Ctrl-Shift-Enter.
Matrix multiplication has some of the same properties as "normal" multiplication, such as
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APPENDIX Econometrics I – Matrix Algebra
Ing. Martina Hanová, PhD.
However matrix multiplication is not communicative. That is to say A*B does not necessarily
equal B*A. In fact, B*A often has no meaning since the dimensions rarely match up.
The INVERSE of a square matrix A, sometimes called a reciprocal matrix, is a matrix A-1. The
matrix A-1 is the INVERSE of matrix A if
A*A-1 = A-1*A = I
where I is the identity matrix. A square matrix A has an inverse if the determinant |A|≠ 0.
=
5
10
6
2
12
4
3
8
1
&
0.28
= 0.53
0.44
0.19
0.32
0.11
0.72
0.97
0.56
For finding the matrix inverse in general, you can use Gauss-Jordan Algorithm. However, this
is a rather complicated algorithm, so usually one relies upon the computer or calculator to
find the matrix inverse.
EXCEL: If A is an n × n square array, then MINVERSE(A) = A-1. This is an array function and so
you must highlight an n × n range before entering =MINVERSE(A) and then pressing CtrlShift-Enter.
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