+
−.. 4.3 LO: Matrices (3) [anoop] 
+
−.. -----------------------------
+
−.. * creating matrices 
+
−..   + direct data 
+
−..   + list conversion 
+
−..   + builtins - identitiy, zeros, 
+
−.. * matrix operations 
+
−..   + + - * / 
+
−..   + dot 
+
−..   + inv 
+
−..   + det 
+
−..   + eig 
+
−..   + norm 
+
−..   + svd 
+
−
+
−========
+
−Matrices
+
−========
+
−{{{ show the welcome slide }}}
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−
+
−Welcome to the spoken tutorial on Matrices.
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−
+
−{{{ switch to next slide, outline slide }}}
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−
+
−In this tutorial we will learn about matrices, creating matrices and
+
−matrix operations.
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−
+
−{{{ creating a matrix }}}
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−
+
−All matrix operations are done using arrays. Thus all the operations
+
−on arrays are valid on matrices also. A matrix may be created as,
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−::
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−
+
−    m1 = matrix([1,2,3,4])
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−
+
−Using the tuple ``m1.shape`` we can find out the shape or size of the
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−matrix,
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−::
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−
+
−    m1.shape
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−
+
−Since it is a one row four column matrix it returned a tuple, one by
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−four.
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−
+
−A list can be converted to a matrix as follows,
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−::
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−
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−    l1 = [[1,2,3,4],[5,6,7,8]]
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−    m2 = matrix(l1)
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−
+
−Note that all matrix operations are done using arrays, so a matrix may
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−also be created as
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−::
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−
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−    m3 = array([[5,6,7,8],[9,10,11,12]])
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−
+
−{{{ switch to next slide, matrix operations }}}
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−
+
−We can do matrix addition and subtraction as,
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−::
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−
+
−    m3 + m2
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−
+
−does element by element addition, thus matrix addition.
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−
+
−Similarly,
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−::
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−
+
−    m3 - m2
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−
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−it does matrix subtraction, that is element by element
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−subtraction. Now let us try,
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−::
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−
+
−    m3 * m2
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−
+
−Note that in arrays ``array(A) star array(B)`` does element wise
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−multiplication and not matrix multiplication, but unlike arrays, the
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−operation ``matrix(A) star matrix(B)`` does matrix multiplication and
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−not element wise multiplication. And in this case since the sizes are
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−not compatible for multiplication it returned an error.
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−
+
−And element wise multiplication in matrices are done using the
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−function ``multiply()``
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−::
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−
+
−    multiply(m3,m2)
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−
+
−Now let us see an example for matrix multiplication. For doing matrix
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−multiplication we need to have two matrices of the order n by m and m
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−by r and the resulting matrix will be of the order n by r. Thus let us
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−first create two matrices which are compatible for multiplication.
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−::
+
−
+
−    m1.shape
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−
+
−matrix m1 is of the shape one by four, let us create another one of
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−the order four by two,
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−::
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−
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−    m4 = matrix([[1,2],[3,4],[5,6],[7,8]])
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−    m1 * m4
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−
+
−thus unlike in array object ``star`` can be used for matrix multiplication
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−in matrix object.
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−
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−{{{ switch to next slide, recall from arrays }}}
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−
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−As we already saw in arrays, the functions ``identity()``,
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−``zeros()``, ``zeros_like()``, ``ones()``, ``ones_like()`` may also be
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−used with matrices.
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−
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−{{{ switch to next slide, matrix operations }}}
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−
+
−To find out the transpose of a matrix we can do,
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−::
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−
+
−    print m4
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−    m4.T
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−
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−Matrix name dot capital T will give the transpose of a matrix
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−
+
−{{{ switch to next slide, Euclidean norm of inverse of matrix }}}
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−
+
−Now let us try to find out the Euclidean norm of inverse of a 4 by 4
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−matrix, the matrix being,
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−::
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−
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−    m5 = matrix(arange(1,17).reshape(4,4))
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−    print m5
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−
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−The inverse of a matrix A, A raise to minus one is also called the
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−reciprocal matrix such that A multiplied by A inverse will give 1. The
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−Euclidean norm or the Frobenius norm of a matrix is defined as square
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−root of sum of squares of elements in the matrix. Pause here and try
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−to solve the problem yourself, the inverse of a matrix can be found
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−using the function ``inv(A)``.
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−
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−And here is the solution, first let us find the inverse of matrix m5.
+
−::
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−
+
−    im5 = inv(m5)
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−
+
−And the euclidean norm of the matrix ``im5`` can be found out as,
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−::
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−
+
−    sum = 0
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−    for each in array(im5.flatten())[0]:
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−        sum += each * each
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−    print sqrt(sum)
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−
+
−{{{ switch to next slide, infinity norm }}}
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−
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−Now try to find out the infinity norm of the matrix im5. The infinity
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−norm of a matrix is defined as the maximum value of sum of the
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−absolute of elements in each row. Pause here and try to solve the
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−problem yourself.
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−
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−The solution for the problem is,
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−::
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−
+
−    sum_rows = []
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−    for i in im5:
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−        sum_rows.append(abs(i).sum())
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−    print max(sum_rows)
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−
+
−{{{ switch to slide the ``norm()`` method }}}
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−
+
−Well! to find the Euclidean norm and Infinity norm we have an even easier
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−method, and let us see that now.
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−
+
−The norm of a matrix can be found out using the method
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−``norm()``. Inorder to find out the Euclidean norm of the matrix im5,
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−we do,
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−::
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−
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−    norm(im5)
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−
+
−And to find out the Infinity norm of the matrix im5, we do,
+
−::
+
−
+
−    norm(im5,ord=inf)
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−
+
−This is easier when compared to the code we wrote. Do ``norm``
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−question mark to read up more about ord and the possible type of norms
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−the norm function produces.
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−
+
−{{{ switch to next slide, determinant }}}
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−
+
−Now let us find out the determinant of a the matrix m5. 
+
−
+
−The determinant of a square matrix can be obtained using the function
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−``det()`` and the determinant of m5 can be found out as,
+
−::
+
−
+
−    det(m5)
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−
+
−{{{ switch to next slide, eigen vectors and eigen values }}}
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−
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−The eigen values and eigen vector of a square matrix can be computed
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−using the function ``eig()`` and ``eigvals()``.
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−
+
−Let us find out the eigen values and eigen vectors of the matrix
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−m5. We can do it as,
+
−::
+
−
+
−    eig(m5)
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−
+
−Note that it returned a tuple of two matrices. The first element in
+
−the tuple are the eigen values and the second element in the tuple are
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−the eigen vectors. Thus the eigen values are,
+
−::
+
−
+
−    eig(m5)[0]
+
−
+
−and the eigen vectors are,
+
−::
+
−
+
−    eig(m5)[1]
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−
+
−The eigen values can also be computed using the function ``eigvals()`` as,
+
−::
+
−
+
−    eigvals(m5)
+
−
+
−{{{ switch to next slide, singular value decomposition }}}
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−
+
−Now let us learn how to do the singular value decomposition or S V D
+
−of a matrix.
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−
+
−Suppose M is an m�n matrix whose entries come from the field K, which
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−is either the field of real numbers or the field of complex
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−numbers. Then there exists a factorization of the form
+
−
+
−    M = U\Sigma V star
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−
+
−where U is an (m by m) unitary matrix over K, the matrix \Sigma is an
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−(m by n) diagonal matrix with nonnegative real numbers on the
+
−diagonal, and V*, an (n by n) unitary matrix over K, denotes the
+
−conjugate transpose of V. Such a factorization is called the
+
−singular-value decomposition of M.
+
−
+
−The SVD of matrix m5 can be found as
+
−::
+
−
+
−    svd(m5)
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−
+
−Notice that it returned a tuple of 3 elements. The first one U the
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−next one Sigma and the third one V star.
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−    
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−{{{ switch to next slide, recap slide }}}
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−
+
−So this brings us to the end of this tutorial. In this tutorial, we
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−learned about matrices, creating matrices, matrix operations, inverse
+
−of matrices, determinant, norm, eigen values and vectors and singular
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−value decomposition of matrices.
+
−
+
−{{{ switch to next slide, thank you }}}
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−
+
−Thank you!
+
−
+
−..  Author: Anoop Jacob Thomas <anoop@fossee.in>
+
−    Reviewer 1:
+
−    Reviewer 2:
+
−    External reviewer: