Matrices

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Chapter 6.1

• Matrices

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• Issue organizing information.
• A medical study researches the relation between
  height and weight. A number of people are asked
  to provide their data, anonymously, and the
  results are entered in a table, which might look
  as follows (height is in inches and weight in
  pounds)

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• All the data now is present, and perfectly
  organized if one needs the heights, one looks at
  the first column for weights, the second column
  if one is interested in finding whether a person
  has certain measurements, one looks at the rows.
  It is not only the number that is important, but
  also what it represents.

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• We notice the following characteristics of our
  data
• all persons have both height and weight
• no person has more than these two
• no person has some other data (age, for example)
• As a result, data can be organized in a
  rectangular table

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• Because of its rectangular shape, this kind of
  table can be characterized by its width and
  length, namely the number of rows and the
  number of columns it has (in our example 3 rows,
  2 columns.)
• We call such a table a 3x2 matrix (three by two
  matrix)
• 3x2 is called the size, or order, of the matrix

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• Of course, we can imagine that such a table has
  more than just 3 rows, or less more than 2
  columns, or less. As long as the table stays
  rectangular, its still going to be a matrix. The
  difference is going to be in its dimensions

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• A 1x4 matrix
• a 4x1 matrix

• A 2x3 matrix
• a 3x3 matrix

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• Notice two peculiarities
• brackets and parentheses ( ) are used to
  confine the matrix (useful when one writes many
  matrices next to each other) brackets are more
  popular, so we are going to use them from now on
• rows are always mentioned first, and columns
  second hence, a 3x2 matrix has 3 rows and 2
  columns, and NOT the other way around

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• We can now generalize the situation for two
  natural numbers m and n, a mxn matrix is a table
  that has m rows and n columns, and looks like
  follows

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• A few words about the previous table
• each a stands for a number
• the indices should be viewed as pairs (hence the
  upper-leftmost a has index one-one, rather than
  eleven)
• the first index stands for the row in which that
  number is, while the second stands for the column
  (the rule row first - column second is
  preserved)

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• What about those strange dots? Its a common
  trick, to symbolize those many missing numbers
  (which, since we dont know what values m and n
  have, it would be impossible to put down anyway)
• the general element of a matrix is usually
  denoted by
• and stands for the element at the inter-section
  of row i with column j

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• A usual method of constructing a matrix is to use
  the indices to compute the element corresponding
  to a certain row and certain column - useful for
  when we dont want to write down the whole
  matrix, but rather present the formula for its
  elements
• Example

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• stands for

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• Matrices come in various sizes (orders), as you
  can see - but two kinds deserve special
  attention
• 1xn matrices, called row matrices, or row vectors
  (why? because they only have 1 row)
• mx1 matrices, called column matrices, or column
  vectors
• Why are they important? Because they are the
  building blocks of all matrices

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• (for instance, a 3x2 matrix is made out of 2
  column vectors OR out of 3 row vectors)
• a matrix with one row and one column (that is,
  has a single element) is both row and column
  vector

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• Some matrices are important not because of their
  size, but rather because of the data they contain
  (usually mostly 0)
• the zero matrix, or 0 (a bigger, bold zero sign),
  which is a matrix with all its elements zero
  (matrices of any size)
• diagonal matrices all elements are zero, with
  the exception of those elements whose indices are
  11, 22, 33, , etc (square sizes matrices only)

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• (notice that if the matrix has square size - such
  a matrix is called a square matrix - those
  elements constitute the diagonal of the square
  represented by the matrix)
• upper triangular matrices all elements below the
  diagonal are zero lower triangular matrices all
  elements above the diagonal are zero (again, only
  square matrices)
• Exercise give examples of those special matrices

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• When experimenting, sometimes it is imperative
  that we obtain same results all the time. So,
  when the data used is organized as matrices, how
  can we make sure that two such matrices are
  equal?
• The idea they have to be identical - not only
  the data must be the same, but also the
  positions must coincide

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• Hence, we have a two step process in establishing
  whether a matrix is equal to another
• step one check their sizes (orders) - must have
  same number of rows and same number of columns
• step two check each element against its
  counterpart (for example, the element of index 11
  in the first matrix must be identical to the
  element 11 in the second matrix, etc)

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• Examples
• not equal, since they have different orders (even
  though its the same data!)
• same order, but not same positioning (the
  element of index 11 in the first is not same as
  element of index 11 in the second and again,
  same data is involved)

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• equality, since they are identical (same order,
  same positioning)

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• Finally, an interesting issue the first table we
  discussed, we considered it natural to have the
  height and weight on columns, and the persons on
  each row what if we considered it otherwise?
  Have the persons correspond to columns, and the
  height and weight as rows? It would be a
  different matrix, but the information is not
  altered, nor its representation

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• We call such a change transposition (or, one
  matrix is the others transposed) and is denoted
  by a capital T in the upper right
• Examples

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• Notice that the change does the following
• the first row becomes the first column
• the second row becomes the second column
• etc

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• A row matrix (vector) becomes a column matrix
  (vector) and viceversa
• the size, in general, changes can you tell for
  what kind of matrices it will not?

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Chapter 6.2

• Matrix Addition and Scalar Multiplication

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• Since matrices have numbers as components, it is
  a natural question whether it is possible to
  combine matrices using operations such as
  addition, subtraction, multiplication etc

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• Lets consider addition and subtraction
• the natural way of defining those operations is
  element by element, namely
• add element 11 of first matrix to element 11 of
  second matrix
• add element 12 of first matrix to element 12 of
  second matrix
• etc

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• (same with subtraction)
• Example

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• Finally, another rather simple operation is
  scaling of a matrix, or, how its called,
  multiplication by a scalar
• given a number (the scalar) and a matrix,
  multiplying the number by the matrix stands for
  multiplying ALL elements of the matrix by the
  number (scale the matrix by the given number)

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• Example

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Chapter 6.3

• Matrix Multiplication

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• Review problems
• for the matrix above give the entries
• that are equal to 3

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• Perform addition, if possible, for the following
  pairs of matrices

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• What is the transposed of the following matrix

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• Matrix addition is quite simple - add
  corresponding entries in two matrices
• one idea for multiplication use same kind of
  rule!
• Unfortunately, not natural, nor rich enough

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• Idea behind matrix multiplication go to the
  building blocks of matrices, namely row
  matrices and column matrices
• define the following multiplication rule between
  a row matrix and a column matrix

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• Notice a few things about this definition
• the row matrix has 4 entries, hence 4 columns
• the column matrix has 4 entries as well, which
  correspond to 4 rows
• the result is a matrix with only one entry, a
  number basically

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• Examples

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• Counterexample
• does not work!!! Must have same number of entries

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• Lets start making things more complex, by using
  instead of just a column matrix, a matrix with
  TWO columns

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• If you remember, we can rewrite the matrix on the
  right so it shows how its made out of columns

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• Now use the same definition as before for each of
  the columns

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• Example

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• And, of course, we can complicate things even
  further, by using instead of a single row matrix,
  a two rows matrix

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• Again, the trick is to view the left matrix as
  made out of rows, and the right matrix as made
  out of columns

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• The result

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• Awfully complicated, right?
• In fact, not quite - lets analyze the situation
  a bit
• in order to be able to multiply the matrices
  using our rule we need to have the rows in the
  first matrix have same number of entries as the
  columns in the second matrix

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• But what does that mean? The number of entries in
  the rows of the first matrix is THE NUMBER OF
  COLUMNS of the first matrix
• the number of entries in the columns of the
  second matrix is THE NUMBER OF ROWS of the second
  matrix
• in conclusion we can only multiply matrices that
  have this above property satisfied - and it can
  be seen right away if one looks at the sizes

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• Example a 2x3 matrix can be multiplied by a 3x5
  matrix a 7x20 matrix can be multiplied by a 20x2
  matrix a 3x3 matrix CANNOT be multiplied by a
  2x3 matrix (for all the cases above, in that
  order
• observation order is VERY important
• lets continue - what can you say about the
  dimension of the resulting matrix?
• The number of rows is same as number of rows of
  first matrix

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• The number of columns is same as number of
  columns of second matrix
• in conclusion the inner dimension tells you
  whether the multiplication can happen, but then
  gets lost, and only the outside dimensions are
  left
• Example a 3x5 matrix multiplied by a 5x7 matrix
  5 is same, so they can be multiplied, while the
  resulting matrix has dimensions 3x7

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• How about the entries of the resulting matrix? We
  had those strange formulas above but they all
  follow a logical pattern (lets look back at the
  above product) we have a 2x3 matrix multiplied
  by a 3x2 matrix the result must be a 2x2 matrix
  (the common 3 tells us the product works, and
  then just disappears) lets say we want the
  entry on the first row, second column - it is
  produced by the first row of the first matrix
  times the second column of the second matrix

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• The entries in the first row of the first matrix
  are so entries which have
  first index 1, and any second index
• the entries in the second column of the second
  matrix are so entries
  which have second index 2, and any first index
• these get to be multiplied respectively
• what do we have?

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• Now we can think of what happens in the general
  case, cant we? How should the two matrices look
  - they should be of sizes mxn the first one, and
  the second one must be nxp (the common n allows
  multiplication, m and p can be anything) the
  result will be a mxp matrix
• entries of the resulting matrix, say a general
  entry,

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• This entry comes from the i-th row of the first
  times the j-th column of the second
• the i-th row of the first matrix has the
  following entries
• the j-th column of the second matrix has the
  following entries
• notice again the importance of the same n

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• Finally, the ij entry of the result is
• notice how each term has outside indices i and
  j, while the inner index is same for both
  factors that make up that term

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• Observations
• not any two matrices can be multiplied together
• matrix multiplication has similar properties as
  number multiplication for example, it is
  associative (a product of three can be broken
  down ABC(AB)CA(BC)) distributive with respect
  to addition (one can distribute a matrix
  multiplied with a sum of other two matrices
  A(BC)ABAC)

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• (Counter)-observation
• as mentioned, multiplication is not commutative,
  not even in the case it is possible (for square
  matrices, in fact)
• yet, scaling (the multiplication by a scalar)
  commutes, and does commute within a product
  (scaling a product is same as scaling the first
  factor and then multiplying, or scaling the
  second one and then multiplying
• k(AB)(kA)BA(kB)

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• Observation
• multiplication is not commutative, yet, using
  transposition, we get some kind of commutativity
• (the transposed of the product is the product of
  the transposed matrices, but in the reverse order)

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• Observation (square matrices)
• if we multiply square matrices, say a nxn by a
  nxn, the result is also nxn - nice feature
• for square matrices we also have the 1, the so
  called unit matrix, or identity matrix, which
  is a diagonal matrix (remember, these only
  existed if the matrix was supposed to be square!)
  with 1 on the diagonal

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• Question for matrix addition do we have a matrix
  which has the role of 0? The answer is YES the
  zero matrix, the 0
• does this zero matrix have same effect as the
  zero for numbers, when used in multiplication?
  YES (if multiplication is possible)

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• Finally what are the real uses of this matrix
  multiplication? Mainly its about writing certain
  relations in a more compact form
• Example
• Alice buys 3 gallons of milk, priced at 1.5 -
  hence pays 3 x 1.54.5
• Bob buys 3 gallons of milk, priced at 1.5 AND 2
  loaves of bread, .99 each - how much does he pay?

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• Of course, 3 x 1.5 2 x 0.99 6.48 - but
  its too long more compact
• its compact in the sense of a single
  multiplication being used (at least apparently)

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• Another example - systems of equations

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• Notice something interesting we keep using the
  same x, y, z for all the rows how about we
  write those unknowns as a matrix, and try to get
  each row of the system as a matrix product.
• First row

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• Second row
• third row

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• Looks very much like what we did in the beginning
  so lets consolidate all these
• the resulting matrix should be ...

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• Hence we can write the system as follows

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• This approach is going to be very useful
  because we will start discussing matrices
  properties - and we will have a very powerful
  tool for solving even monster systems of equations

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• Solutions for the review problems
• for
• 3 is in position 12 (first row, second column)
  and in position 21(second row, forst column)

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