Matrix

Basics

Identity Matrix

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

$IA=A\!$

For any m-by-n matrix $A$.

For example the identity matrix in R ³

$\mathbf {I} _{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}$

Vectors

A three diemensional vector

${\overrightarrow {v}}_{3}=v_{1}{\hat {i}}+v_{2}{\hat {j}}+v_{3}{\hat {k}}\!$

has the matrix representation

${\overrightarrow {v}}_{3}={\begin{bmatrix}v_{1}\\v_{2}\\v_{3}\end{bmatrix}}$

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

${\overrightarrow {v}}_{n}={\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}$

Determinants

The determinant of a 2-by-2 matrix

$A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}$

is

$det{\mathit {A}}={\begin{vmatrix}a&b\\c&d\\\end{vmatrix}}=ad-bc$

Eigenvalue Analysis

Let

${\mathit {A}}={\begin{bmatrix}2&3\\3&-6\end{bmatrix}}$

We must find all scalars $\lambda$ such that the matrix equation

$(A-\lambda I)\mathbf {x} =\mathbf {0}$

so we subtract $A$ by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda I}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A - \lambda I = \begin{bmatrix} 2 & 3 \\ 3 & -6 \end{bmatrix} - \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} = \begin{bmatrix} 2-\lambda & 3 \\ 3 & -6 - \lambda \end{bmatrix} }

So the eigenvalues of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} are the solutions of the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle det(A - \lambda I) = \begin{vmatrix} 2-\lambda & 3 \\ 3 & -6-\lambda \end{vmatrix} = 0 }

This gives us

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2 - \lambda)(-6 - \lambda) - (3)(3) = \lambda^2 + 4\lambda - 21 = 0\!}

Solving this polynomial we find that the eigenvalues of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = 3, -7\!}