Vectors and matrices

Element-wise against matrix arithmetic, indexing, and the aggregates.

v

[\begin{matrix} 3 \\ 4 \\ 12 \end{matrix}]

w

[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]

Between two vectors of equal length, * is element-wise: what a tabulated calculation wants, column against column.

elementwise

v \cdot w

[3, 4, 12] \cdot [1, 2, 3]

=[3, 8, 36]

An inner product is asked for by name, so it is never confused with the above.

inner

dot (v, w)

dot ([3, 4, 12], [1, 2, 3])

47

Scalars broadcast.

scaled

2 \cdot v

2 \cdot [3, 4, 12]

=[6, 8, 24]

Aggregates.

total

sum (v)

sum ([3, 4, 12])

19

smallest

\min (v)

\min ([3, 4, 12])

3

largest

\max (v)

\max ([3, 4, 12])

12

n

length (v)

length ([3, 4, 12])

3

Indexing is one-based.

second

v_{2}

{([3, 4, 12])}_{2}

4

A vector is the column of n that rows and cols say it is, so it takes the column index as well: these are the same element. Any other column is out of bounds, which is what a column of one means.

{also}_{second}

v_{2, 1}

{([3, 4, 12])}_{2, 1}

4

Vectors carry units element by element, so a tabulated calculation keeps them.

f

[\begin{matrix} 5 \\ 10 \\ 15 \end{matrix}] Hz

acc

\frac{[\begin{matrix} 2.89 \\ 3.04 \\ 3.72 \end{matrix}] m}{s^{2}}

stroke

\frac{acc}{{(2 \cdot π \cdot f)}^{2}}

\frac{[2.89 m·s⁻², 3.04 m·s⁻², 3.72 m·s⁻²]}{{(2 \cdot π \cdot [5 Hz, 10 Hz, 15 Hz])}^{2}}

=[0.00292818 m, 0.000770041 m, 0.000418794 m]

stroke

=[2.92818 mm, 0.770041 mm, 0.418794 mm]

A matrix is a vector of rows, and is indexed row first.

K

[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]

k21

K_{2, 1}

{([[1, 2], [3, 4]])}_{2, 1}

3

Between two matrices * is the matrix product, not element-wise. identity(n) writes the n×n one, which is dimensionless so that S - λ*identity(2) — the characteristic equation of a stress tensor — takes its dimension from λ.

I

identity (2)

=[[1, 0], [0, 1]]

product

K \cdot I

[[1, 2], [3, 4]] \cdot [[1, 0], [0, 1]]

=[[1, 2], [3, 4]]

diag writes a matrix with a vector down its diagonal — a mass matrix, here. The zeros carry the diagonal's dimension so the result is usable in arithmetic.

mass

diag ([\begin{matrix} 3 \\ 4 \end{matrix}] kg)

=[[3 kg, 0 kg], [0 kg, 4 kg]]

Linear algebra.

d

\det (K)

\det ([[1, 2], [3, 4]])

-2

Kt

transpose (K)

transpose ([[1, 2], [3, 4]])

=[[1, 3], [2, 4]]

Kinv

inv (K)

inv ([[1, 2], [3, 4]])

=[[-2, 1], [1.5, -0.5]]

The check that inv is really an inverse.

{back}_{to_I}

K \cdot Kinv

[[1, 2], [3, 4]] \cdot [[-2, 1], [1.5, -0.5]]

=[[1, -1.11022e-16], [8.88178e-16, 1]]

The cross product is three-dimensional, and units come out of the products it is made of: metres crossed with newtons give newton-metres. -> N*m asks for the moment to be shown that way rather than in joules, which share the dimension but not the meaning.

r

[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] m

F

[\begin{matrix} 0 \\ 2 \\ 0 \end{matrix}] N

M

cross (r, F)

cross ([1 m, 0 m, 0 m], [0 N, 2 N, 0 N])

=[0 N*m, 0 N*m, 2 N*m]

Parallel vectors cross to zero, which is the check worth writing down.

parallel

cross (r, r)

cross ([1 m, 0 m, 0 m], [1 m, 0 m, 0 m])

=[0 m², 0 m², 0 m²]

Shape and slicing. A vector answers as the column it is.

n

rows (K)

rows ([[1, 2], [3, 4]])

2

second

col (K, 2)

col ([[1, 2], [3, 4]], 2)

=[2, 4]

Joining: side by side, and one above the other.

side

augment ([\begin{matrix} 1 \\ 2 \end{matrix}] m, [\begin{matrix} 3 \\ 4 \end{matrix}] m)

=[[1 m, 3 m], [2 m, 4 m]]

piled

stack ([\begin{matrix} 1 \\ 2 \end{matrix}] m, [\begin{matrix} 3 \\ 4 \end{matrix}] m)

=[1 m, 2 m, 3 m, 4 m]