Matrix

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Basics

Identity Matrix

The identity matrix, , is defined as the matrix that satisfies the condition

For any m-by-n matrix .

For example the identity matrix in R 3


Vectors

A three diemensional vector

has the matrix representation

Or more generally, an n-diemensional vector has the matrix form

Addition & Subtraction

Only matrices with the same dimensions can be added and subtracted. If we take two matrices with dimensions , and , then we will get a resultant matrix, , with entries

Similarly, for subtraction

Determinants

The determinant of a 2-by-2 matrix

is

Eigenvalue Analysis

Let

We must find all scalars such that the matrix equation

so we subtract by

So the eigenvalues of are the solutions of the equation

This gives us

Solving this polynomial we find that the eigenvalues of are