Linear Algebra

Giac.jl provides comprehensive symbolic linear algebra capabilities including matrix operations, determinants, eigenvalues, and linear system solving.

Setup

using Giac, LinearAlgebra
using Giac.Commands: eigenvalues, linsolve

@giac_var a b c d x y

Matrix Creation

Numeric Matrices

Create matrices from Julia arrays:

A = GiacMatrix([1 2; 3 4])
# 2×2 GiacMatrix:
#  1  2
#  3  4

Symbolic Matrices

Create matrices with symbolic entries:

@giac_var a b c d
B = GiacMatrix([[a, b],
                [c, d]])
# 2×2 GiacMatrix:
#  a  b
#  c  d

Symbol Constructor

Create large symbolic matrices using the symbol constructor:

@giac_several_vars m 2 2
M = GiacMatrix([[m11, m12],
                [m21, m22]])
det(M)  # m11*m22-m12*m21

# Or use the compact constructor:
M = GiacMatrix(:m, 3, 3)
# 3×3 GiacMatrix with entries m11, m12, ...

For very large matrices:

M = GiacMatrix(:m, 100, 100)
# 100×100 GiacMatrix:
#   m_1_1    m_1_2    m_1_3    ...    m_1_100
#   m_2_1    m_2_2    m_2_3    ...    m_2_100
#      ⋮        ⋮        ⋮     ⋱          ⋮
# m_100_1  m_100_2  m_100_3   ...  m_100_100

Custom Index Ranges

Create matrices with non-1-based indexing using UnitRange or StepRange arguments:

# 0-based indexing (useful for physics, quantum mechanics)
M = GiacMatrix(:ψ, 0:2, 0:2)
# 3×3 GiacMatrix with entries ψ00, ψ01, ψ02, ψ10, ψ11, ...

# Negative indices (useful for angular momentum states)
J = GiacMatrix(:J, -1:1)
# 3×1 column vector with entries J_m1, J_0, J_1 (m = minus)

# Custom ranges
A = GiacMatrix(:A, 5:7, 1:3)
# 3×3 GiacMatrix with entries A51, A52, A53, A61, ...

# StepRange for even indices
E = GiacMatrix(:E, 0:2:6)
# 4×1 column vector with entries E0, E2, E4, E6

# Mixed integer and range arguments
M = GiacMatrix(:M, 3, 0:2)
# 3×3 matrix: rows use 1:3, columns use 0:2
# Entries: M10, M11, M12, M20, M21, M22, M30, M31, M32

The same range syntax works with @giac_several_vars:

@giac_several_vars ψ 0:2
# Creates: ψ0, ψ1, ψ2

@giac_several_vars T 0:1 0:2
# Creates: T00, T01, T02, T10, T11, T12

@giac_several_vars c -1:1
# Creates: c_m1, c_0, c_1 (m = minus for negative indices)

Naming Convention:

• Indices 0-9: concatenated directly (e.g., m12, ψ00)
• Indices > 9: underscore separators (e.g., m_1_10)
• Negative indices: m prefix for minus (e.g., -1 → m1, so c_m1 for index -1)

Determinant

Compute the determinant using det:

A = GiacMatrix([1 2; 3 4])
det(A)
# Output: -2

# Symbolic determinant
@giac_var a b c d
B = GiacMatrix([[a, b], [c, d]])
det(B)
# Output: a*d-b*c

Inverse

Compute the matrix inverse using inv:

A = GiacMatrix([1 2; 3 4])
inv(A)
# Returns the inverse matrix

For singular matrices, GIAC will return an error.

Trace

Compute the trace (sum of diagonal elements) using tr:

A = GiacMatrix([1 2; 3 4])
tr(A)
# Output: 5

Transpose

Compute the transpose using transpose:

A = GiacMatrix([1 2; 3 4])
transpose(A)
# 2×2 GiacMatrix:
#  1  3
#  2  4

Eigenvalues and Eigenvectors

Eigenvalues

Compute eigenvalues using the eigenvalues command from Giac.Commands:

using Giac.Commands: eigenvalues

A = GiacMatrix([2 1; 1 2])
# Convert GiacMatrix to GiacExpr via ptr for eigenvalues command
eigenvalues(GiacExpr(A.ptr))
# Output: 3,1

Characteristic Polynomial

The characteristic polynomial can be computed for eigenvalue analysis.

Linear System Solving

Using linsolve

Solve systems of linear equations using linsolve:

using Giac.Commands: linsolve

@giac_var x y

# System: x + y = 3, x - y = 1
linsolve([x + y ~ 3, x - y ~ 1], [x, y])
# Output: [2, 1]

Using solve

The general solve command also works for linear systems:

using Giac.Commands: solve

solve([x + y ~ 3, x - y ~ 1], [x, y])
# Returns the solution

Matrix Rank

Compute the rank of a matrix using invoke_cmd(:rank, ...):

A = GiacMatrix([1 2 3; 4 5 6; 7 8 9])
# Use invoke_cmd because rank conflicts with LinearAlgebra.rank
# Convert GiacMatrix to GiacExpr via ptr
invoke_cmd(:rank, GiacExpr(A.ptr))
# Output: 2 (linearly dependent rows)

Matrix Operations

Addition and Subtraction

A = GiacMatrix([1 2; 3 4])
B = GiacMatrix([5 6; 7 8])

A + B
# 2×2 GiacMatrix:
#  6   8
# 10  12

A - B
# 2×2 GiacMatrix:
# -4  -4
# -4  -4

Matrix Multiplication

A = GiacMatrix([1 2; 3 4])
B = GiacMatrix([5 6; 7 8])

A * B
# 2×2 GiacMatrix:
# 19  22
# 43  50

Scalar Multiplication

A = GiacMatrix([1 2; 3 4])

2 * A
# 2×2 GiacMatrix:
# 2  4
# 6  8

Matrix Powers

A = GiacMatrix([1 2; 3 4])

A^2
# Computes A * A

Table of Functions

Notes

• All matrix operations work symbolically
• For numerical evaluation, use to_julia to convert results
• Large symbolic matrices can be created efficiently using the symbol constructor
• The ~ operator creates equations for use with linsolve and solve