.. _vectors: Vectors and Matrices in MATLAB ==================================== .. include:: ../replace.txt .. index:: scalar, vector, matrix, column vector, row vector, transpose .. index:: isscalar, isvector, isempty, ismatrix, iscolumn, isrow .. only:: html .. topic:: Reading Assignment Please continue reading chapter 4 of *Physical Modeling in MATLAB* from section 4.5, by Allen B. Downey [DOWNEY11]_. |M| (named for MATrix LABoratory) is designed to work with variables that actually contain multiple numeric values. This allows us to perform calculations on every value in the variable with just one expression. It also allows |M| programs to conveniently solve problems using linear algebra. We will put off the linear algebra aspect for now and focus on *vectorized* equations that can do calculations on many values at once without the need to write a ``for`` loop. The numeric variables that we have used so far have only one value. In math terminology, we call such variables *scalars*. |M| reports that the size of scalars are 1 x 1. We will now introduce vectors and matrices, but focus on vectors. Vectors can also be called *arrays* for common terminology with other programming languages, but note that |M| has both row vectors and column vectors, which differs from most other programming languages. .. describe:: scalar A variable with size 1 x 1. That is, it has only one value. See `isscalar function `_. .. describe:: row vector A variable of size :math:`1{\times}n`, where :math:`n > 1`. It has one row containing :math:`n` values. Row vectors can be created with the colon operator, manually entered, or created with one of several |M| functions. See `isrow function `_, and `isvector function `_. What we call a row vector in |M| terminology often holds data that could be called an array. :: >> C = 1:6 % C is a 1x6 row vector C = 1 2 3 4 5 6 >> D = [3 5 7 9] % D is a 1x4 row vector D = 3 5 7 9 .. describe:: column vector A variable of size :math:`m{\times}1`, where :math:`m > 1`. It has one column containing :math:`m` values. Column vectors can be created with the transpose of a row vector, manually entered, or created with one of several |M| functions. See `iscolumn function `_. Column vectors may hold data and also often contain the coordinates for geometric points and vectors. The transpose is an operation that switches the rows and columns of either a vector or a matrix. :: >> E = (1:6)' % ' is the transpose operator E = 1 2 3 4 5 6 >> F = [3; 5; 7; 9] F = 3 5 7 9 .. describe:: matrix A variable of size :math:`m{\times}n`, where usually :math:`m > 1` and :math:`n > 1`. It has :math:`m` rows and :math:`n` columns of data values. Matrices can be manually entered, or created with one of several |M| functions. See `ismatrix function `_. Note that vectors (:math:`m = 1` or :math:`n = 1`) are considered matrices; although, we will attempt to distinguish between them since vectors have some unique properties. Matrices may also be formed by concatenating row vectors or column vectors. :: >> G = [1 2 3; 4 5 6; 7 8 9] G = 1 2 3 4 5 6 7 8 9 >> A = [1 2 4]; >> B = [2 1 6]; >> H = [A; B] H = 1 2 4 2 1 6 >> D = [1; 1; 2]; >> E = [0; 1; 3]; >> J = [D E] J = 1 0 1 1 2 3 The command [ [A B]; C ] will combine the following matrices as shown. .. figure:: matrixCombine.png :align: center .. describe:: Higher dimension data Data arrays containing more than two dimensions are also possible. Color images have size :math:`m{\times}3` It contains three monochrome images for each red, blue, and green. .. seealso:: `isempty function `_ tests if an array is empty or contains data. .. note:: Vectors in our physical 3-dimensional world are said to be in :math:`\mathbb{R}^3`, that is they consist of three real numbers defining their magnitude in the :math:`\bf{x}`, :math:`\bf{y}`, and :math:`\bf{z}` directions. Vectors on a plane, like a piece of paper, are said to be in :math:`\mathbb{R}^2`. Vectors may also be used for applications not relating to geometry and may have higher dimension than 3. Generally, we call this :math:`\mathbb{R}^n`. For some applications, the coefficients of vectors may also be complex numbers, which is denoted as :math:`\mathbb{C}^n`. Row vectors are usually used for sequences of numbers, such as is used for plotting. However, when referring to a geometric vector in either :math:`\mathbb{R}^2` or :math:`\mathbb{R}^3` space, then the term *vector* refers to a column vector. For convenience of notation, the elements of a column vector may be written as a comma separated row of numbers between parenthesis. These are both column vectors in :math:`\mathbb{R}^3`. .. math:: \spalignvector{1; 2; 3} = (1, 2, 3) Matrix Generating Functions ---------------------------- .. index:: ones, zeros, preallocating storage One example of when it is desired to create a simple matrix or vector of pre-determined size is when a loop is used to set or modify the vector data. For example, consider the following lines of code:: for k=1:5 A(k) = k; end To this code, |M| gives a warning that says: The variable 'A' appears to change size with each loop iteration (within a script). Consider preallocating for speed. The warning is fixed by first creating the vector as follows:: A = zeros(1,5); for k=1:5 A(k) = k; end |M| has several functions that will create matrices and vectors. The ``zeros`` and ``ones`` functions are commonly used. More information can be found `in the MATLAB documentation `_. :: >> A = zeros(1,5) A = 0 0 0 0 0 >> B = ones(2) B = 1 1 1 1 >> C = ones(3,1) C = 1 1 1 >> D = zeros(3,2) D = 0 0 0 0 0 0 Scalar - Vector Arithmetic --------------------------- .. index:: vector arithmetic When an arithmetic operation occurs between a scalar and a vector or matrix, the operation is applied to each element of the vector. :: >> A = zeros(1,5) A = 0 0 0 0 0 >> A = A + 2 A = 2 2 2 2 2 >> A = A*3 A = 6 6 6 6 6 >> A = A/2 A = 3 3 3 3 3 >> A = A - 1 A = 2 2 2 2 2 .. _element-wise: Element-wise Arithmetic --------------------------- .. index:: element-wise operators, .*, ./, .^ Addition and subtraction arithmetic between vectors and matrices happens as expected as long as the two matrices are the same size. :: >> A = 2*ones(1,4) A = 2 2 2 2 >> B = ones(1,4) B = 1 1 1 1 >> A + B ans = 3 3 3 3 When we want to perform multiplication, division, or an exponent *element-wise*, that is between each element of the vectors, we use the ``.*``, ``.^`` and ``./`` element-wise operators. These operators take two vectors or matrices of the same size and perform the operation between pairs of values at the same position in the vectors or matrices. .. describe:: .* :: >> a = [2 3 4]; >> b = [5 6 7]; >> c = a.*b c = 10 18 28 .. describe:: ./ :: >> c ./ a ans = 5 6 7 .. describe:: .^ The element-wise exponent can be used between vectors and between a scalar and a vector. :: >> 2.^a ans = 4 8 16 >> a.^2 ans = 4 9 16 >> b.^a ans = 25 216 2401 .. note:: Matrix and vector multiplication (``A*B``) and division (``A\B`` and ``A/B``) is covered under the :ref:`linAlg` section. Vector and Matrix Indices ------------------------------ .. index:: vector indices, indices You can select elements of a vector with parentheses. The range of valid indices for a vector of length N is 1 to N. .. warning:: Most other programming languages count indices from 0 to N-1. Just remember that |M| is different in this regard. |M| produces an error when a vector (array) is indexed outside of its range. However, when an assignment statement adds elements beyond the current range then the vector is expanded. :: >> A = [2 4 6 8] A = 2 4 6 8 >> A(1) ans = 2 >> A(4) ans = 8 >> A(5) Index exceeds the number of array elements (4). >> A(5) = 10 A = 2 4 6 8 10 Ranges of Indices ^^^^^^^^^^^^^^^^^^ .. index:: end The colon operator can be used to select a subset of a vector. |M| keeps a special identifier called ``end`` for the last element of a vector. Ranges of indices can be used when reading from a vector and when assigning values to it. :: >> A = 1:10; >> A(1:5) ans = 1 2 3 4 5 >> A(6:end) = A(1:5) A = 1 2 3 4 5 1 2 3 4 5 >> A(2:2:end-2) = [10 9 8 7] A = 1 10 3 9 5 8 2 7 4 5 .. _arrayIndex: Accessing Data in Matrices ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. index:: matrix indices You can extract values from an array using row, column indexing. :: >> x = MatrixData(rowNum,colNum); You can use the |M| keyword ``end`` as either a row or column index to reference the last element. The following line of code accesses the last element in `MatrixData`. :: >> x = MatrixData(end, end) x = 92 You can use arithmetic with the keyword ``end``. For example:: >> x = MatrixData(end-1,end-2) x = 72 To extract an entire row or column from an array, you can use `:`. Think of the colon as saying *all rows* or *all columns*. This extracts a column from ``MatrixData``:: >> x = MatrixData(:,colNum); What if you want to extract a range of rows or columns? You can index using a vector containing those index values. For example, both lines of the following code will extract the first three rows of ``MatrixData``. :: >> firstThree = MatrixData(1:3,:); >> firstThree = MatrixData([1 2 3],:); What index values will return all four corners of ``MatrixData``? :: >> x = MatrixData([1 end],[1 end]) Delete Vector or Matrix Data ------------------------------ .. index:: delete data Assigning vector or matrix elements to take values of ``[]`` deletes them. The selection of the elements to delete can use either array indexing or :ref:`logicalVects`. :: >> v = 1:6 v = 1.00 2.00 3.00 4.00 5.00 6.00 >> v(2:4) = [] v = 1.00 5.00 6.00 >> B = randi(20, 1, 5) % random numbers B = 19.00 14.00 16.00 15.00 8.00 >> B(B > 15) = [] % delete based on a logical vector B = 14.00 15.00 8.00 Linear and Logarithmic Spaced Vectors -------------------------------------- .. index:: linspace, logspace An alternative to the colon operator for creating vectors that span a range of number is the ``linspace`` function. The ``linspace`` function returns a row vector with a fixed number of elements. Use ``linspace`` when you want a certain number of elements in the vector or are more concerned with covering a range of values than using specific values. The first and second arguments to ``linspace`` give the range. The third argument, which is optional, gives the number of elements. The default for the third argument is 100, which is a nice value for creating a smooth line plot. The ``logspace`` function also generates a sequence of numbers, but the numbers are evenly spaced on logarithmic scale. The first two arguments to ``logspace`` specify the range as an exponent to 10 (:math:`1 \leadsto 10; 2 \leadsto 100`). Like ``linspace``, the third argument gives the number of elements desired. The default number of elements is 50. :: >> x = linspace(0,10,5) x = 0 2.5000 5.0000 7.5000 10.0000 >> y = linspace(-pi, pi); >> whos Name Size Bytes Class Attributes x 1x5 40 double y 1x100 800 double >> logspace(0,2,5) ans = 1.0000 3.1623 10.0000 31.6228 100.0000 .. raw:: latex \clearpage