Linear algebra

The matrix product

@ always means matmul, never elementwise.

A = [[1.0, 2.0], [3.0, 4.0]]
B = [[0.0, 1.0], [1.0, 0.0]]
show A @ B

transpose

A = [[1.0, 2.0], [3.0, 4.0]]
show transpose(A)

The symmetric eigensolver

eig_sym fills its arguments: values ascending, eigenvectors in columns. Trace = sum of eigenvalues.

A = [[2.0, 1.0], [1.0, 3.0]]
vals = zeros(2)
vecs = zeros(2, 2)
eig_sym(A, vals, vecs)
show vals
show vals[0] + vals[1]

Matrix-vector products via matmul

A 2×2 times a length-2 vector.

A = [[0.0, 1.0], [1.0, 0.0]]
v = [3.0, 7.0]
show A @ v

Chained products

R = [[0.0, 0.0 - 1.0], [1.0, 0.0]]
show R @ R @ R @ R

A 3x3 eigenproblem

The eigenvalues come out in ascending order and sum to the trace, 9.

A = [[2.0, 1.0, 0.0], [1.0, 3.0, 1.0], [0.0, 1.0, 4.0]]
vals = zeros(3)
vecs = zeros(3, 3)
eig_sym(A, vals, vecs)
show vals
show vals[0] + vals[1] + vals[2]

Solvers that return their results

The same solvers as the forms that fill arrays you pass, which still work: the values ascend, and lin_solve's answer gives back b.

A = [[2.0, 1.0, 0.0], [1.0, 3.0, 1.0], [0.0, 1.0, 4.0]]
vals, vecs = eig_sym(A)
show vals
x = lin_solve(A, [1.0, 2.0, 3.0])
show A @ x

transpose reverses the axes

Of a matrix, the rows become columns; of a rank-3 array, all three axes turn around; a vector comes back as it was, as in NumPy. transpose(A, axes) puts them in any order.

T = transpose(reshape(arange(24), 2, 3, 4))
show shape(T)
show transpose([1.0, 2.0])

The dot product is dot, not @

@ is the matrix product and refuses two vectors; dot(u, v) sums u[i] * v[i] left to right, and outer(u, v) is the other product of two vectors. The cell prints 11 from dot, then stops at the @ line.

show dot([1.0, 2.0], [3.0, 4.0])
show [1.0, 2.0] @ [3.0, 4.0]

expected error: `@` of two vectors is refused…