Boyd_Matrix_Primer - A primer on matrices Stephen Boyd...

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A primer on matrices Stephen Boyd September 16, 2008 These notes describe the notation of matrices, the mechanics of matrix manipulation, and how to use matrices to formulate and solve sets of simultaneous linear equations. We won’t cover linear algebra, i.e. , the underlying mathematics of matrices numerical linear algebra, i.e. , the algorithms used to manipulate matrices and solve linear equations software for forming and manipulating matrices, e.g. , Matlab, Mathematica, or Octave how to represent and manipulate matrices, or solve linear equations, in computer lan- guages such as C/C++ or Java applications, for example in statistics, mechanics, economics, circuit analysis, or graph theory 1 Matrix terminology and notation Matrices A matrix is a rectangular array of numbers (also called scalars ), written between square brackets, as in A = 0 1 2 . 3 0 . 1 1 . 3 4 0 . 1 0 4 . 1 1 0 1 . 7 . An important attribute of a matrix is its size or dimensions , i.e. , the numbers of rows and columns . The matrix A above, for example, has 3 rows and 4 columns, so its size is 3 × 4. (Size is always given as rows × columns.) A matrix with m rows and n columns is called an m × n matrix. An m × n matrix is called square if m = n , i.e. , if it has an equal number of rows and columns. Some authors refer to an m × n matrix as fat if m < n (fewer rows than columns), or skinny if m > n (more rows than columns). The matrix A above is fat. 1
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The entries or coefficients of a matrix are the values in the array. The i, j entry is the value in the i th row and j th column, denoted by double subscripts: the i, j entry of a matrix C is denoted C ij (which is a number). The positive integers i and j are called the (row and column, respectively) indices . For our example above, A 13 = 2 . 3, A 32 = 1. The row index of the bottom left entry (which has value 4 . 1) is 3; its column index is 1. Two matrices are equal if they are the same size and all the corresponding entries (which are numbers) are equal. Vectors and scalars A matrix with only one column, i.e. , with size n × 1, is called a column vector or just a vector . Sometimes the size is specified by calling it an n -vector . The entries of a vector are denoted with just one subscript (since the other is 1), as in a 3 . The entries are sometimes called the components of the vector, and the number of rows of a vector is sometimes called its dimension . As an example, v = 1 2 3 . 3 0 . 3 is a 4-vector (or 4 × 1 matrix, or vector of dimension 4); its third component is v 3 = 3 . 3. Similarly, a matrix with only one row, i.e. , with size 1 × n , is called a row vector . As an example, w = bracketleftBig 2 . 1 3 0 bracketrightBig is a row vector (or 1 × 3 matrix); its third component is w 3 = 0.
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