Mathematica Symbolic Matrix and Vector Operations

Mathematica is very strong at symbolic matrix and vector operations. Let us go through the basics step by step.

1. Define vectors and matrices

In Mathematica:

• A vector is a list.
• A matrix is a list of lists.

v = {x, y, z}
A = {{a, b}, {c, d}}

You can inspect dimensions with Dimensions:

Dimensions[v]
Dimensions[A]

2. Matrix operations

• Transpose

Transpose[A]

• Matrix multiplication uses the dot operator .

A.v
A.A

• Hadamard product

A*B

For linear algebra operations, do not use * in place of ..

• Determinant and inverse

Det[A]
Inverse[A]

• Identity matrix

IdentityMatrix[3]

3. Symbolic linear algebra

Mathematica can also handle symbolic linear algebra:

• Eigenvalues and eigenvectors

Eigenvalues[A]
Eigenvectors[A]

• Characteristic polynomial

CharacteristicPolynomial[A, λ]

• Solve a linear system $Ax=b$

Solve[A.{x1, x2} == {1, 0}, {x1, x2}]

4. Basic vector operations

For symbolic vectors:

• Dot product

Dot[v, v]
v.v

• Cross product

Cross[{x1, x2, x3}, {y1, y2, y3}]

• Norm

Norm[v]

5. Better output formatting

You can use MatrixForm to display matrices more clearly:

A // MatrixForm

贡献者: Junyuan He