LG at 15:10, 10 September 2013

2013-09-10T15:10:54Z

LG at 15:02, 10 September 2013

2013-09-10T15:02:55Z

LG: Created page with "= Matrices = In the PreSession Maths course, a matrix was defined as follows:
A matrix is a rectangular array of numbers enclosed in parentheses, con- ventional..."

2013-09-10T14:54:50Z

Created page with "= Matrices = In the PreSession Maths course, a matrix was defined as follows: <blockquote>A matrix is a rectangular array of numbers enclosed in parentheses, con- ventional..."

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@@ Line 644: / Line 644: @@
 a_{12} & a_{22}\\
 a_{13} & a_{23}
-\end{array}\right]</math> and that the <math>\left(3,2\right)</math> element of this product is actually <math>c_{23}</math>: <math>b_{13}a_{21}+b_{23}a_{22}+b_{33}a_{23}=a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}=c_{23}.</math> In summation notation, we see that from <math>B^{T}A^{T}</math>: <math>c_{23}=\sum_{k=1}^{3}b_{k3}a_{2k},</math> where the position of the index of summation is due to the transposition. So, in summation notation, the calculation of <math>c_{23}</math> from <math>B^{T}A^{T}</math> equals that from equation (6).
+\end{array}\right]</math> and that the <math>\left(3,2\right)</math> element of this product is actually <math>c_{23}</math>: <math>b_{13}a_{21}+b_{23}a_{22}+b_{33}a_{23}=a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}=c_{23}.</math> In summation notation, we see that from <math>B^{T}A^{T}</math>: <math>c_{23}=\sum_{k=1}^{3}b_{k3}a_{2k},</math> where the position of the index of summation is due to the transposition. So, in summation notation, the calculation of <math>c_{23}</math> from <math>B^{T}A^{T}</math> equals that from equation (4).
 More generally, the <math>\left(i,j\right)</math> element of <math>AB</math>: <math>\sum_{k=1}^{3}a_{ik}b_{kj}</math> is the <math>\left(j,i\right)</math> element of <math>B^{T}A^{T}.</math> But this means that <math>B^{T}A^{T}</math> must be the transpose of <math>AB,</math> since the elements in the <math>i</math>th row of <math>AB</math> are being written in the <math>i</math>th column of <math>B^{T}A^{T}.</math>
@@ Line 956: / Line 956: @@
 The point about the use of partitioned matrices is that the product <math>A\mathbf{x}</math> can be represented as: <math>A\mathbf{x}=A_{1}\mathbf{x}_{1}+A_{2}\mathbf{x}_{2}+A\mathbf{x}_{3}</math> by applying the across and down rule to the submatrices and the subvectors, a much simpler representation than the use of summations.
-Each of the components is a conformable matrix-vector product: this is essential in any use of partitioned matrices to represent some matrix product. For example, using <math>A</math> from equation (8) and <math>B</math> as: <math>B=\left[\begin{array}{c}
+Each of the components is a conformable matrix-vector product: this is essential in any use of partitioned matrices to represent some matrix product. For example, using <math>A</math> from equation (6) and <math>B</math> as: <math>B=\left[\begin{array}{c}
 B_{11}\\
 B_{21}\\

@@ Line 3: / Line 3: @@
 In the PreSession Maths course, a matrix was defined as follows:
-<blockquote>A matrix is a rectangular array of numbers enclosed in parentheses, con-
+<blockquote>A matrix is a rectangular array of numbers enclosed in parentheses, conventionally denoted by a capital letter. The number of rows (say <math>m</math>) and
 the number of columns (say <math>n</math>) determine the order of the matrix (<math>m</math> <math>\times</math> <math>n</math>).
@@ Line 115: / Line 113: @@
 == Transposition of vectors ==
-The ''rows'' of the matrix <math>A</math> in equation ([eq:axy]) can be seen as elements of column vectors, say:
+The ''rows'' of the matrix <math>A</math> in equation (1) can be seen as elements of column vectors, say:
 <math>\begin{aligned}
@@ Line 274: / Line 272: @@
 \end{array}\right],</math> it is clear that <math>\mathbf{x}^{T}\mathbf{y}=0.</math> This seems a rather innocuous definition, and yet the idea of orthogonality turns out to be extremely important in econometrics.
-If <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are thought of as points in <math>R^{2},</math> and arrows are drawn from the origin to <math>\mathbf{x}</math> and to <math>\mathbf{y,}</math> then the two arrows are perpendicular to each other - see Figure [orthy<sub>e</sub>xample]. If <math>\mathbf{y}</math> were defined as: <math>\mathbf{y}=\left[\begin{array}{r}
+If <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are thought of as points in <math>R^{2},</math> and arrows are drawn from the origin to <math>\mathbf{x}</math> and to <math>\mathbf{y,}</math> then the two arrows are perpendicular to each other - see Figure 1. If <math>\mathbf{y}</math> were defined as: <math>\mathbf{y}=\left[\begin{array}{r}
 \\
 -1
 \end{array}\right],</math> the position of the <math>\mathbf{y}</math> vector and the corresponding arrow would change, but the perpendicularity property would still hold.
-[ht] [[Image:0C__courses_econometric_methods_yed_orthy_example.pdf|fig:]]
+Figure 1
 === Matrix - vector products ===
@@ Line 378: / Line 377: @@
 <math>\begin{aligned}
 B & = & \left[\begin{array}{rrrr}
-\mathbf{b}_{1} & \mathbf{b}_{2} & \ldots & \mathbf{b}_{r}\end{array}\right]\text{\,\,\,\,\,\,\,(by columns)}\\
+\mathbf{b}_{1} & \mathbf{b}_{2} & \ldots & \mathbf{b}_{r}\end{array}\right]\ \ \ \ \ \ \ \text{(by columns)}\\
-  & = & \left\Vert b_{ik}\right\Vert ,\ \ \ \ \ i=1,...,n,k=1,...,r\text{ \,\,\,\,\,\,(typical element)}\\
+  & = & \left\Vert b_{ik}\right\Vert ,\ \ \ \ \ i=1,...,n,k=1,...,r\ \ \ \ \ \ \ \text{(typical element)}\\
   & = & \left[\begin{array}{rrrr}
 b_{11} & b_{12} & \ldots & b_{1r}\\
@@ Line 385: / Line 384: @@
 \vdots & \vdots & \ddots & \vdots\\
 b_{n1} & b_{n2} & \ldots & b_{nr}
-\end{array}\right]\text{\,\,\,\,\,\,\,(the array)}\end{aligned}</math>
+\end{array}\right]\ \ \ \ \ \ \ \text{(the array)}\end{aligned}</math>
-What does the typical element of the <math>m\times r</math> matrix <math>C</math> look like? Start with the <math>k</math>th column of <math>C,</math> which is <math>A\mathbf{b}_{k}.</math> The <math>i</math>th element in <math>A\mathbf{b}_{k}</math> is, from equation ([eq:ab]), the inner product of the elements of the <math>i</math>th row in <math>A</math>: <math>\left[\begin{array}{rrrr}
+What does the typical element of the <math>m\times r</math> matrix <math>C</math> look like? Start with the <math>k</math>th column of <math>C,</math> which is <math>A\mathbf{b}_{k}.</math> The <math>i</math>th element in <math>A\mathbf{b}_{k}</math> is, from equation (2), the inner product of the elements of the <math>i</math>th row in <math>A</math>: <math>\left[\begin{array}{rrrr}
 a_{i1} & a_{i2} & \ldots & a_{in}\end{array}\right],</math> with the elements of <math>\mathbf{b}_{k},</math> so that the inner product is: <math>a_{i1}b_{1k}+a_{i2}b_{2k}+\ldots+a_{in}b_{nk}=\sum_{j=1}^{n}a_{ij}b_{jk}.</math>
@@ Line 645: / Line 644: @@
 a_{12} & a_{22}\\
 a_{13} & a_{23}
-\end{array}\right]</math> and that the <math>\left(3,2\right)</math> element of this product is actually <math>c_{23}</math>: <math>b_{13}a_{21}+b_{23}a_{22}+b_{33}a_{23}=a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}=c_{23}.</math> In summation notation, we see that from <math>B^{T}A^{T}</math>: <math>c_{23}=\sum_{k=1}^{3}b_{k3}a_{2k},</math> where the position of the index of summation is due to the transposition. So, in summation notation, the calculation of <math>c_{23}</math> from <math>B^{T}A^{T}</math> equals that from equation ([eq:c23]).
+\end{array}\right]</math> and that the <math>\left(3,2\right)</math> element of this product is actually <math>c_{23}</math>: <math>b_{13}a_{21}+b_{23}a_{22}+b_{33}a_{23}=a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}=c_{23}.</math> In summation notation, we see that from <math>B^{T}A^{T}</math>: <math>c_{23}=\sum_{k=1}^{3}b_{k3}a_{2k},</math> where the position of the index of summation is due to the transposition. So, in summation notation, the calculation of <math>c_{23}</math> from <math>B^{T}A^{T}</math> equals that from equation (6).
 More generally, the <math>\left(i,j\right)</math> element of <math>AB</math>: <math>\sum_{k=1}^{3}a_{ik}b_{kj}</math> is the <math>\left(j,i\right)</math> element of <math>B^{T}A^{T}.</math> But this means that <math>B^{T}A^{T}</math> must be the transpose of <math>AB,</math> since the elements in the <math>i</math>th row of <math>AB</math> are being written in the <math>i</math>th column of <math>B^{T}A^{T}.</math>
@@ Line 957: / Line 956: @@
 The point about the use of partitioned matrices is that the product <math>A\mathbf{x}</math> can be represented as: <math>A\mathbf{x}=A_{1}\mathbf{x}_{1}+A_{2}\mathbf{x}_{2}+A\mathbf{x}_{3}</math> by applying the across and down rule to the submatrices and the subvectors, a much simpler representation than the use of summations.
-Each of the components is a conformable matrix-vector product: this is essential in any use of partitioned matrices to represent some matrix product. For example, using <math>A</math> from equation ([eq:partition<sub>a</sub>]) and <math>B</math> as: <math>B=\left[\begin{array}{c}
+Each of the components is a conformable matrix-vector product: this is essential in any use of partitioned matrices to represent some matrix product. For example, using <math>A</math> from equation (8) and <math>B</math> as: <math>B=\left[\begin{array}{c}
 B_{11}\\
 B_{21}\\

Lnotes - Revision history

LG at 15:10, 10 September 2013

LG at 15:02, 10 September 2013

LG: Created page with "= Matrices = In the PreSession Maths course, a matrix was defined as follows: A matrix is a rectangular array of numbers enclosed in parentheses, con- ventional..."

LG: Created page with "= Matrices = In the PreSession Maths course, a matrix was defined as follows:
A matrix is a rectangular array of numbers enclosed in parentheses, con- ventional..."