Matrix functions

The functions that work on vectors and matrices, one entry each. For writing a matrix, indexing it and the bars |M|, see Matrices; the table functions, which read a matrix as a table, are on Table functions, and everything else on Built-in functions.

column

column(M, j)

One column of a matrix, counting from 1.

Parameters

Parameter	Meaning
M	A matrix with rows and columns. A table block’s name works too. See Tables.
j	The column number, from 1.

Returns

The column, as an $n\times 1$ matrix.

Errors

Message	When
Only two dimensional matrix is supported	M is a plain list rather than a matrix with rows and columns
Index out of range	j is below 1 or past the last column
Index must be an integer	j isn’t a whole number

Example

$$\mathrm{column}\left(\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],2\right)=\left[\begin{array}{c}2\\ 4\end{array}\right]$$

cross

cross(a, b)

The cross product of two three-dimensional vectors.

Parameters

Parameter	Meaning
a	A vector of exactly three elements.
b	A vector of exactly three elements.

Returns

The vector $a\times b$.

Errors

Message	When
Vectors with length 3 expected	Either vector doesn’t have three elements
Too few arguments in function cross / Too many arguments in function cross	Not exactly two arguments

Example

$$\mathrm{cross}\left(\left[\begin{array}{ccc}1& 2& 3\end{array}\right],\left[\begin{array}{ccc}4& 5& 6\end{array}\right]\right)=\left[\begin{array}{ccc}-3& 6& -3\end{array}\right]$$

det

det(M)

The determinant of a square matrix. The bars |M| around a square matrix mean the same thing.

Parameters

Parameter	Meaning
M	A square matrix.

Returns

A number.

Errors

Message	When
Matrix must be square	M isn’t square, or is a plain list
Too few arguments in function det / Too many arguments in function det	Not exactly one argument

Example

$$\begin{array}{rl}\det \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]& =-2\\ \det \left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 10\end{array}\right]& =-3\end{array}$$

dot

dot(a, b)

The dot product of two vectors of the same length. Units multiply through.

Parameters

Parameter	Meaning
a	A vector: a plain list or a column matrix.
b	A vector of the same length.

Returns

A number, or a quantity whose units are the product of the two vectors’ units.

Errors

Message	When
Vectors must have equal length	The two vectors differ in length
Too few arguments in function dot / Too many arguments in function dot	Not exactly two arguments

Example

$$\begin{array}{rl}\mathrm{dot}\left(\left[\begin{array}{ccc}1& 2& 3\end{array}\right],\left[\begin{array}{ccc}4& 5& 6\end{array}\right]\right)& =32\\ \mathrm{dot}\left(\left[\begin{array}{cc}1\mathrm{m}& 2\mathrm{m}\end{array}\right],\left[\begin{array}{cc}3\mathrm{N}& 4\mathrm{N}\end{array}\right]\right)& =11\mathrm{J}\end{array}$$

inv

inv(M)

The inverse of a square matrix. Of a plain number, its reciprocal.

Parameters

Parameter	Meaning
M	A square, non-singular matrix, or a number.

Returns

The inverse matrix, or $1/x$ for a number.

Errors

Message	When
Cannot calculate inverse, determinant is zero	M is singular
Matrix must be square	M isn’t square
Too few arguments in function inv / Too many arguments in function inv	Not exactly one argument

Example

$$\mathrm{inv}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}-2& 1\\ 1.5& -0.5\end{array}\right]$$

norm

norm(v)
norm(v, p)
norm(M)

The length of a vector, the everyday case, including a vector of quantities, as long as they share a unit. Of a matrix, the Frobenius norm; of a number, its magnitude. The bars |v| around a vector mean the same thing, and also accept a column written as a matrix, which norm itself does not.

Parameters

Parameter	Meaning
v	A vector written as a plain list, of numbers or of quantities with one unit.
p	Optional. The kind of norm: 2 (default, Euclidean), 1 (sum of magnitudes), Infinity (largest magnitude), or 'fro'.
M	A matrix with rows and columns.

Returns

The length, in the unit of the first element; a number for plain values.

Errors

Message	When
The length of a vector needs every element in the same unit	Some elements carry a unit and others don’t, or the units differ in kind
Invalid matrix dimensions	A column vector written as an $n\times 1$ matrix. Use the bars, or a plain list
Too few arguments in function norm / Too many arguments in function norm	No argument, or more than two

Example

$$\begin{array}{rl}\mathrm{norm}\left(\left[\begin{array}{cc}3& 4\end{array}\right]\right)& =5\\ \mathrm{norm}\left(\left[\begin{array}{cc}3\mathrm{mm}& 4\mathrm{cm}\end{array}\right]\right)& =40.1123\mathrm{mm}\\ \mathrm{norm}\left(\left[\begin{array}{ccc}1& 2& 3\end{array}\right],\infty \right)& =3\end{array}$$

row

row(M, i)

One row of a matrix, counting from 1.

Parameters

Parameter	Meaning
M	A matrix with rows and columns. A table block’s name works too. See Tables.
i	The row number, from 1.

Returns

The row, as a $1\times n$ matrix.

Errors

Message	When
Only two dimensional matrix is supported	M is a plain list rather than a matrix with rows and columns
Index out of range	i is below 1 or past the last row
Index must be an integer	i isn’t a whole number

Example

$$\mathrm{row}\left(\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],2\right)=\left[\begin{array}{cc}3& 4\end{array}\right]$$

transpose

transpose(M)

Swaps a matrix’s rows and columns. The toolbar’s $M^{\mathrm{T}}$ is the same function. A plain list has no second dimension, so it comes back unchanged; write a column as a matrix to transpose it into a row.

Parameters

Parameter	Meaning
M	A matrix.

Returns

The transposed matrix.

Errors

Message	When
Too few arguments in function transpose / Too many arguments in function transpose	Not exactly one argument

Example

$$\mathrm{transpose}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$$