Difference between revisions of "Matrix (MathObject Class)"

Matrix Class

The Matrix class implements matrices (in any dimension) of Real entries, though the most common use is 2-dimensional ones (i.e., $n\times m$ matrices). A 1-dimensional Matrix is effectively a Vector. Typically, the rows of a Matrix are comma-separated lists of numbers delimited by square brackets, and a Matrix is a comma separated list of rows delimited by square brackets. E.g., [[1,2,3],[4,5,6]] is a $2\times 3$ matrix whose first row has entries 1, 2 and 3, and whose second row has entries 4, 5 and 6. Matrices can be formed in the Matrix Context. It is possible to have Complex entries in a Matrix, but there is not a pre-defined Context that makes that easy to do.

The answer checker for Matrices can give students hints about whether the dimension of the matrix is correct. It is also possible to have an array of answer blanks rather than having the student type the entire matrix in one long answer blank.

Construction

Matrices are created via the Matrix() function, or by Compute().

Context("Matrix");

$M = Matrix([1,2,3],[4,5,6]);$M = Matrix([[1,2,3],[4,5,6]]);
$M = Matrix("[[1,2,3],[4,5,6]]");$M = Compute("[[1,2,3],[4,5,6]]");

Higher-dimensional Matrices can be obtained as a comma-separated list of lower-dimensional matrices delimited by square brackets. For example,

$M = Matrix([[[1,2],[3,4]],[[5,6],[7,8]]]) produces a $2\times 2\times 2$ matrix (a 3-dimensional matrix). The identity matrix for any dimension can be obtained from Value::Matrix->I(n) where n is the dimension of the matrix. For example$I = Value::Matrix->I(2);

is a $2\times 2$ identity matrix.

Operations on Matrices

Matrices can be combined by addition, subtraction, and multiplication, when the dimensions match up properly. Square matrices can have integer powers taken of them (meaning repeated multiplication). You can perform scalar multiplication and division, and also matrix-vector multiplication.

$M1 = Matrix([1,2],[3,4]);$M2 = Matrix([5,6],[7,8]);
$v = Vector(9,10);$M3 = $M1 +$M2;     # same as Matrix([6,8],[10,12]);
$A =$M1 * $M2; # usual matrix multiplication$B = $M1 ** 2; # same as$M1 * $M1$C = 3 * $M1; # same as Matrix([3,6],[9,12]);$D = $M1 / 2; # same as Matrix([.5,1],[1.5,2]);$u = $M1 *$v        # matrix-vector multiplication

The value of $u above will be a $2\times 1$ matrix, which is effectively a column vector but it is not a Vector object, it is a Matrix object. You can force it to be a Vector object using$u = Vector($M1 *$v); or $u = ColumnVector($m1 * $v) instead. Matrices have a number of methods that can be used to compute things like determinants, inverses, transposes, and so on; these are listed in the Methods section below. Some examples are:$a = $M1->det;$A = $M1->transpose;$B = $M1->inverse;$t = $M1->trace;$v = $M1->row(1); # the first row of the matrix$v = $M1->column(1); # the first column of the matrix$a = $M1->element(2,1); # the element in the second row, first column Note that$M1->row(1) and $M1->column(1) produce Matrix objects, not Vectors. You can coerce them to Vectors via Vector($M1->row(1)) and ColumnVector($M1->column(1)). Answer Checker As with all MathObjects, you obtain an answer checker for a Matrix object via the cmp() method: ANS(Matrix([[4,0,-2],[1,1,5]])->cmp); The Matrix class supports the common answer-checker options, and the following additional options: Option Description Default showDimensionHints => 1 or 0 Show/don't show messages about the wrong number of rows or columns. 1 If you use a single answer blank for a Matrix answer checker, this means the student must type the entire Matrix, including the brackets and commas, all on one line. This allows the student to enter Matrix-values expressions (like sums or multiplications), but it is hard to type, and difficult to see the Matrix as an array when it is presented linearly in this way. It is more common, therefore, to use the Matrix object's ans_array() method rather than PG's ans_rule() function to obtain a separate answer box for each entry. The Matrix answer checker will manage this collection of answer boxes for you, so your code doesn't have to change in any other way. Context("Matrix");$M = Matrix([[2,4,1],[-1,0,3]]);

Context()->texStrings;
BEGIN_TEXT
$$M$$ = \{$M->ans_array\} END_TEXT Context()->normalStrings; ANS($M->cmp);

If you want students to enter Matrix-valued expressions, you can add Matrix-valued constants to the Context of your problem and have them write answers as matrix expressions in terms of those constants. For example

Context("Numeric");        # Numeric prevents students from entering Matrices or Vectors directly

M => Matrix([pi**2,1/pi],[sqrt(2),ln(pi)]),    # an arbitrary matrix with no special properties or symmetries
I => Matrix([[1,0],[0,1]]),                    # identity matrix
);
Context()->flags->set(
);

$f = Formula("M^2 + 3M - I"); # a Matrix formula Context()->texStrings; BEGIN_TEXT $$f$$ = \{ans_rule(20)\} END_TEXT Context()->normalStrings; ANS($f->cmp);

Methods

The Matrix class supports the Common MathObject Methods, and the following additional methods:

Method Description
$M->det Returns the determinant of$M as a Real.
$M->inverse Returns the inverse of$M. Note that $M must be square.$M->transpose Returns the transpose of $M.$M->trace Returns the trace of $M. Note that$M must be square.
$M->proj($V) Returns the projection of $V onto the column space of$M. Note that $V must be a Matrix with one column.$M->proj_coeff($V) Returns the coordinates of$M->proj($V). As with$M->proj, the columns of $M should be linearly independent.$M->row(n) Returns the $n$-th row of $M as a 1-dimensional Matrix. To obtain it as a Vector, use Vector($M->row(n)).
$M->column(n) Returns the $n$-th column of$M as a 1-dimensional Matrix. To obtain it as a Vector, use Vector($M->column(n)) or ColumnVector($m->column(n)).
$M->element(i,j) Returns the $(i,j)$-th element of$M (i.e., the element in row $i$ and column $j$) as a Real.
$M->dimensions Returns an array of number of rows and columns of$M if it is 2-dimensional. (If 1-dimensional, it is the number of elements in $M, if 3-dimensional, it is three number giving the sizes of the three dimension, and so on.)$M->isZero Returns 1 if the matrix is a zero matrix and 0 otherwise. (Overrides MathObject method.)
$M->isOne Returns 1 if the matrix is an identity matrix and 0 otherwise. (Overrides MathObject method.)$M->isSquare Returns 1 if the matrix is $n\times n$ and 0 otherwise.
$M->isRow Returns 1 if the matrix is 1-dimensional, 0 otherwise.$M->is_symmetric Returns 1 if the matrix is symmetric across the main diagonal, 0 otherwise.
$M->wwMatrix Returns returns an old-style Matrix from the Matrix package. Note that if$M1 is an old-style Matrix object, Matrix($M1) will convert it to a MathObject Matrix.$M->norm_one Returns the one-norm of $M (the maximum of the sums of the columns).$M->norm_max Returns the maximum-norm of $M (the maximum of the sums of the rows).$M->kleene Returns a copy of $M where Kleene's algorithm has been applied. Note that$M must be square.
$M->normalize($v) Returns a matrix and vector that have been "normalized" to improve numeric stability, but still represent the same system as the original $M and$v. Note that $M must be square.$M->decompose_LR Returns a matrix that encodes the L-R decomposition of $M. Note that$M must be square. ($L$ is a lower-triangular matrix and $R$ is an upper-triangular matrix where $L R = M$, but the matrix returned here is a special combination of the two stored in one square matrix, the so-called LR matrix.)
$M->L Returns the $L$ matrix from the L-R decomposition of$M.
$M->R Returns the $R$ matrix from the L-R decomposition of$M.
$M->PL Returns the left pivot permutation matrix for the L-R decomposition of$M. (Note that $PL \, L \, R \, PR = M$.)
$M->PR Returns the right pivot permuation matrix for the L-R decomposition of$M. (Note that $PL \, L \, R \, PR = M$.)
$A->solve_LR($b) Solves the system $A\boldsymbol{x}=\boldsymbol{b}$ using an LR matrix and returns an array consisting of the dimension of the solution space, the solution vector (as a Matrix object), and the base matrix (see the MatrixReal1 documentation for details).
$M->condition($Minv) Shorthand for $M->norm_one*$Minv->norm_one. Note that $Minv should be the inverse of$M, and both must be square. The number is a measure of the numerical stability of the matrix; the smaller the number, the better stabiity.
$M->order_LR Returns the order of$M (the number of linearly independent columns or rows).
$A->solve_GSM($x0,$b,$epsilon) Solves the system $A\boldsymbol{x}=\boldsymbol{b}$ using the Global Step Method fixed-point iterative approach. Here, $x0 is a starting guess solution (that should be near the actual solution),$b is the vector $\boldsymbol{b}$, and $epsilon is an error tolerance for determining when to stop. See the MatrixReal1 documentation for details.$A->solve_SSM($x0,$b,$epsilon) Solves the system $A\boldsymbol{x}=\boldsymbol{b}$ using the Single Step Method fixed-point iterative approach. Here,$x0 is a starting guess solution (that should be near the actual solution), $b is the vector $\boldsymbol{b}$, and$epsilon is an error tolerance for determining when to stop. See the MatrixReal1 documentation for details.
$A->solve_RM($x0,$b,$weight,$epsilon) Solves the system $A\boldsymbol{x}=\boldsymbol{b}$ using the Relaxation Method fixed-point iterative approach. Here,$x0 is a starting guess solution (that should be near the actual solution), $b is the vector $\boldsymbol{b}$,$weight is a number between 0 and 2, and $epsilon is an error tolerance for determining when to stop. See the MatrixReal1 documentation for details. Properties The Matrix class supports the Common MathObject Properties, and the following additional ones: Property Description Default$r->{open} The symbol to use for the open square bracket [
\$r->{close} The symbol to use for the close square bracket ]