03. Matrix Equation

Matrices can be used as first-class objects.

The algebra rules are as follows:

  • (A+B)+C=A+(B+C)
  • A+B=B+A
  • A+0=A
  • ??

Multiplication commutativity does NOT hold.

For example let A be a matrix, λ an unknown scalar, v an unknown vector here:

A⋅v⃗=λ⋅v⃗

A trivial and uninteresting solution would be v⃗=0.

The other solutions can be found by rewriting. First notice that there is a 0 vector:

A⋅v⃗-λ⋅v⃗=0⃗

Then promote every term to be a matrix:

A⋅v⃗-λ⋅I⋅v⃗=0⃗

Then notice distributivity:

(A-λ⋅I)⋅v⃗=0⃗

So either A-λ⋅I=0⃗ or v⃗=0⃗ or both.

The more interesting one is the first:

A-λ⋅I=0

Adjoint Matrices (Conjugate Transpose)

The conjugate transpose, Hermitian transpose, or adjoint matrix is the matrix A^H for the matrix A for which holds:

⟨A⋅x⃗,y⃗⟩=⟨x⃗,A^H⋅y⃗⟩ for any vector x, any vector y

To calculate the adjoint matrix A^H, transpose it and conjugate the elements:

A^H:=\overline A^T

Other interesting properties are:

(A+B)^H=A^H+B^H
(r⋅A)^H={\overline r}⋅A^H
(A⋅B)^H=B^H⋅A^H
(A^H)^H=A
(A^H)^{-1}=(A^{-1})^H

The eigenvalues of A^H are the complex conjugates of the eigenvalues of A.

Author: Danny (remove the ".nospam" to send)

Last modification on: Sat, 04 May 2024 .