Editing Transformation matrix (section)

==Finding the matrix of a transformation==

If one has a linear transformation <math>T(x)</math> in functional form, it is easy to determine the transformation matrix ''A'' by transforming each of the vectors of the [[standard basis]] by ''T'', then inserting the result into the columns of a matrix. In other words,
<math display="block">A = \begin{bmatrix} T( \mathbf e_1 ) & T( \mathbf e_2 ) & \cdots & T( \mathbf e_n ) \end{bmatrix}</math>

For example, the function <math>T(x) = 5x</math> is a linear transformation.  Applying the above process (suppose that ''n'' = 2 in this case) reveals that:
<math display="block">T( \mathbf{x} ) = 5 \mathbf{x} = 5I\mathbf{x} = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \mathbf{x}</math>

The matrix representation of vectors and operators depends on the chosen basis; a [[matrix similarity|similar]] matrix will result from an alternate basis. Nevertheless, the method to find the components remains the same.

To elaborate, vector <math>\mathbf v</math> [[linear combination|can be represented]] in basis vectors, <math>E = \begin{bmatrix}\mathbf e_1 & \mathbf e_2 & \cdots & \mathbf e_n\end{bmatrix}</math> with coordinates <math> [\mathbf v]_E = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix}^\mathrm{T}</math>:
<math display="block">\mathbf v = v_1 \mathbf e_1 + v_2 \mathbf e_2 + \cdots + v_n \mathbf e_n = \sum_i v_i \mathbf e_i = E [\mathbf v]_E</math>

Now, express the result of the transformation matrix ''A'' upon <math>\mathbf v</math>, in the given basis:
<math display="block">\begin{align}
  A(\mathbf v)
    &= A \left(\sum_i v_i \mathbf e_i \right)
     = \sum_i {v_i A(\mathbf e_i)} \\
    &= \begin{bmatrix}A(\mathbf e_1) & A(\mathbf e_2) & \cdots & A(\mathbf e_n)\end{bmatrix} [\mathbf v]_E
     = A \cdot [\mathbf v]_E \\[3pt]
    &= \begin{bmatrix}\mathbf e_1 & \mathbf e_2 & \cdots & \mathbf e_n \end{bmatrix}
       \begin{bmatrix}
         a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\
         a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\
          \vdots &  \vdots & \ddots &  \vdots \\
         a_{n,1} & a_{n,2} & \cdots & a_{n,n} \\
       \end{bmatrix}
       \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n\end{bmatrix} 
\end{align}</math>

The <math>a_{i,j}</math> elements of matrix ''A'' are determined for a given basis ''E'' by applying ''A'' to every <math>\mathbf e_j = \begin{bmatrix} 0 & 0 & \cdots & (v_j=1) & \cdots & 0 \end{bmatrix}^\mathrm{T}</math>, and observing the response vector
<math display="block">A \mathbf e_j = a_{1,j} \mathbf e_1 + a_{2,j} \mathbf e_2 + \cdots + a_{n,j} \mathbf e_n = \sum_i a_{i,j} \mathbf e_i.</math>

This equation defines the wanted elements, <math>a_{i,j}</math>, of ''j''-th column of the matrix ''A''.<ref>{{cite book |last= Nearing |first= James |year=2010 |title= Mathematical Tools for Physics |url = http://www.physics.miami.edu/nearing/mathmethods |chapter = Chapter 7.3 Examples of Operators |chapter-url = http://www.physics.miami.edu/~nearing/mathmethods/operators.pdf |access-date = January 1, 2012 |isbn = 978-0486482125 }}</ref>

===Eigenbasis and diagonal matrix===
{{Main|Diagonal matrix|Eigenvalues and eigenvectors}}

Yet, there is a special basis for an operator in which the components form a [[diagonal matrix]] and, thus, multiplication complexity reduces to {{mvar|n}}. Being diagonal means that all coefficients <math>a_{i,j} </math> except <math>a_{i,i}</math> are zeros leaving only one term in the sum <math display="inline">\sum a_{i,j} \mathbf e_i</math> above. The surviving diagonal elements, <math>a_{i,i}</math>, are known as '''eigenvalues''' and designated with <math>\lambda_i</math> in the defining equation, which reduces to <math>A \mathbf e_i = \lambda_i \mathbf e_i</math>. The resulting equation is known as '''eigenvalue equation'''.<ref>{{cite book |last = Nearing |first = James |year = 2010 |title = Mathematical Tools for Physics |url = http://www.physics.miami.edu/nearing/mathmethods |chapter = Chapter 7.9: Eigenvalues and Eigenvectors |chapter-url = http://www.physics.miami.edu/~nearing/mathmethods/operators.pdf |access-date = January 1, 2012 |isbn = 978-0486482125 }}</ref> The [[Eigenvalues and eigenvectors|eigenvectors and eigenvalues are derived from it via the '''characteristic polynomial''']].

With [[Diagonalizable matrix#Diagonalization|diagonalization]], it is [[diagonalizability|often possible]] to [[change of basis|translate]] to and from eigenbases.