Operators in the Student[LinearAlgebra] Package - Maple Programming Help

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Operators in the Student[LinearAlgebra] Package

 Calling Sequence A + B A . B x * A A * x A ^ n A ^ + Transpose(A) A ^ * HermitianTranspose(A) v &x w

Parameters

 A - Matrix, Vector, or scalar B - Matrix, Vector, or scalar x - scalar n - integer v - 3-D Vector w - 3-D Vector

Description

 • To add two Matrices, two Vectors, or a Matrix and a scalar, use the syntax $A+B$.  In the case where one of $A$ or $B$ is a Matrix and the other is a scalar, the scalar is interpreted as a diagonal Matrix of appropriate dimensions, with that scalar value along the diagonal.
 • To multiply two Matrices, a Matrix and a Vector, a Matrix or Vector and a scalar, or to compute the dot product of two Vectors, use the syntax $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B$.  This "." operator is non-commutative, so it does not rearrange the orders of non-scalar terms.
 If $A$ and $B$ are numbers (including complex and extended numerics such as infinity and undefined), then $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B=AB$, that is, the '.' operator extends the '*' operator in this case.
 If $A$ and $B$ are Vectors with the same orientation (that is, both are row Vectors or both are column Vectors) and dimension, then $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B$ is computed as their dot product.
 If one of $A$ and $B$ is a Matrix or a Vector, and the other is a Matrix, Vector, or constant and the previous case does not apply, then their product is computed as the relevant algebraic operation, without reordering.  That is, the '.' operator implements non-commutative multiplication.
 The '.' operator is n-ary, meaning that expressions such as $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}C$ are interpreted as expected.
 Note:  In Maple,  '.' can be interpreted as a decimal point (for example, $3.7$), as part of a range operator (for example, $x..y$), or as the (non-commutative) multiplication operator.  To distinguish between these three circumstances, Maple uses the following rule.
 Any dot that is not part of a range operator (more than one '.' in a row) and not part of a number is interpreted as the non-commutative multiplication operator.
 Note that the interpretation of the phrase "not part of a number" depends on whether you are using 1-D or 2-D input mode.  In 1-D input mode, interpretation proceeds from left to right, and a dot following a number will be interpreted as a decimal point unless that number already contains a decimal point.  In 2-D input mode, interpretation is carried out on the expression as a whole, and because spaces and juxtaposition can be interpreted as multiplication, a dot which is immediately preceded or followed by a number is always interpreted as a decimal point.
 For example, in 1-D input mode, 3.4 is a number, 3. 4 is an error and 3 .4 and 3 . 4 return 12.  3. .4 is 12. and 3..4 is a range.
 In 2-D input mode, 3.4 is a number, 3. 4 and 3 .4 are errors and 3 . 4 returns 12.  3. .4 is an error and 3..4 is again a range.  (All of the errors shown by these examples are due to the rule that a number cannot appear as the right-hand operand of an implicit multiplication operation. In such cases, use of explicit multiplication (*) can avoid the error.  See also 2-D Math Details for more information.)
 • It is an error to use the "*" operator between operands that are both Matrices or Vectors. To multiply a scalar, x, and a Matrix or Vector, you can use $x$ * $A$ or $A$ * $x$.  There is a small, but important, difference between $x\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A$ and $xA$ in this case, where $x$ is a scalar and $A$ is a Matrix or Vector.  Namely, if $x$ is a symbolic expression (not a constant), then $xA$ performs the componentwise multiplication, while $x\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A$ returns unevaluated.
 • If $A$ is a square Matrix, then its (integer) powers can be computed using the syntax ${A}^{n}$, where $n$ is an integer.  If $n$ is negative, ${A}^{n}={\left({A}^{\left(-1\right)}\right)}^{\left(-n\right)}$, where ${A}^{\left(-1\right)}$ is the inverse of $A$ (if it exists).
 • The transpose of a Matrix or Vector is obtained by the special syntax ${A}^{+}$ or the Transpose(A) command.
 Similarly, the Hermitian transpose of a Matrix or Vector is obtained by the special syntax ${A}^{*}$ or the HermitianTranspose(A) command.
 • The cross product of two 3-D Vectors is computed using the syntax $v&xw$.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):$
 > $A≔⟨⟨1,2⟩|⟨3,4⟩|⟨5,6⟩⟩$
 ${A}{≔}\left[\begin{array}{rrr}{1}& {3}& {5}\\ {2}& {4}& {6}\end{array}\right]$ (1)
 > $A+1$
 $\left[\begin{array}{rrr}{2}& {3}& {5}\\ {2}& {5}& {6}\end{array}\right]$ (2)
 > $B≔⟨⟨a,b⟩|⟨c,d⟩⟩$
 ${B}{≔}\left[\begin{array}{cc}{a}& {c}\\ {b}& {d}\end{array}\right]$ (3)
 > $B\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A$
 $\left[\begin{array}{ccc}{a}{+}{2}{}{c}& {3}{}{a}{+}{4}{}{c}& {5}{}{a}{+}{6}{}{c}\\ {b}{+}{2}{}{d}& {3}{}{b}{+}{4}{}{d}& {5}{}{b}{+}{6}{}{d}\end{array}\right]$ (4)
 > $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B$
 > ${A}^{\mathrm{%T}}$
 $\left[\begin{array}{rr}{1}& {2}\\ {3}& {4}\\ {5}& {6}\end{array}\right]$ (5)
 > ${B}^{-1}$
 $\left[\begin{array}{cc}\frac{{d}}{{a}{}{d}{-}{b}{}{c}}& {-}\frac{{c}}{{a}{}{d}{-}{b}{}{c}}\\ {-}\frac{{b}}{{a}{}{d}{-}{b}{}{c}}& \frac{{a}}{{a}{}{d}{-}{b}{}{c}}\end{array}\right]$ (6)
 > ${B}^{\mathrm{%H}}$
 $\left[\begin{array}{cc}\stackrel{{&conjugate0;}}{{a}}& \stackrel{{&conjugate0;}}{{b}}\\ \stackrel{{&conjugate0;}}{{c}}& \stackrel{{&conjugate0;}}{{d}}\end{array}\right]$ (7)
 > $v≔⟨1,0,-1⟩$
 ${v}{≔}\left[\begin{array}{r}{1}\\ {0}\\ {-}{1}\end{array}\right]$ (8)
 > $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}v$
 $\left[\begin{array}{r}{-}{4}\\ {-}{4}\end{array}\right]$ (9)
 > ${v}^{\mathrm{%T}}$
 $\left[\begin{array}{ccc}{1}& {0}& {-}{1}\end{array}\right]$ (10)
 > $v&x\left(⟨2,3,4⟩\right)$
 $\left[\begin{array}{r}{3}\\ {-}{6}\\ {3}\end{array}\right]$ (11)
 > $v\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}⟨2,3,4⟩$
 ${-}{2}$ (12)
 > $3v$
 $\left[\begin{array}{r}{3}\\ {0}\\ {-}{3}\end{array}\right]$ (13)
 > $xA$
 $\left[\begin{array}{ccc}{x}& {3}{}{x}& {5}{}{x}\\ {2}{}{x}& {4}{}{x}& {6}{}{x}\end{array}\right]$ (14)
 > $x\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A$
 ${x}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{.}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left[\begin{array}{rrr}{1}& {3}& {5}\\ {2}& {4}& {6}\end{array}\right]$ (15)