Linear Algebra

Basic Concepts

Trace

Trace should be calculated using the metric. An example is the trace of Ricci tensor,

R=gabRab

Einstein equation is

Rab12gabR=8πGTab

The trace is

gabRab12gabgabR=8πGgabTabR124R=8πGTR=8πGT

Determinant

Some useful properties of determinant.

  1. Interchange rows (colomns) once will generate a negative sign.
  2. Determinant can be calculated recursively when implemented numerically.
  3. Determinant for block matrix can be expressed using the blocks.

Here is an example of the determinant of block matrix. Suppose our block matrix is

A=(BCDE),

where each block is a square matrix. We calculate the determinant through

Det(A)=Det(BECD).

This is useful when we have a block diagonalized matrix.

Technique

Inverse of a matrix

Many methods to get the inverse of a matrix. Check wikipedia for Invertible matrix.

Adjugate matrix method for example is here.

A1=A|A|

in which, A is the adjugate matrix of A.

Eigenvalues of AA

One can prove that the eigenvalues of any matrix B that can be written as AA are positive semidefinite.

Proof

Suppose the eigenvectors are Vi with corresponding eigenvalues λi, i.e.,

We now construct a number

On one hand, we have

where ViVi0.

On the other hand,

As long as ViVi0, we have

Tensor Product Space

|ϕ1 and |ϕ2 are elements of Hilbert space H1 and H2. Tensor Product of |ϕ1 and |ϕ2 is denoted as |ϕ1|ϕ2. This operation is linear and distributive.

Tensor product space H1H2 is composed of all the linear combinations of all possible tensor products of elements in H1 and H2.

Inner Product

Inner product of two tensor products

(ϕ|1ϕ|2)(|ψ1|ψ2)=(1ϕψ1)(2ϕψ2)

Operators Applied to Tensor Product

Two operators ˆO1 and ˆO2 works on H1 and H2 respectively applied to tensor product

(ˆO1ˆO2)(|ϕ1|ϕ2)=(ˆO1|ϕ1)(ˆO2|ϕ2)

Solving Linear Equations

First of all, write down the augmented matrix for the equation set.

Elementary row operations are allowed on the augmented matrix. Operate on the matrix until one can read out the solutions.