### Matrix calculus: Matrices and coordinate transformations

### Conjugate matrices

If #\alpha# and #\beta# are bases of a finite dimensional vector space #V#, and the matrix #L_\alpha# of a linear map #L: V\to V# with respect to #\alpha# is given, then, according to the theorem *Basis transition*, the matrix #L_\beta# of #L# with respect to #\beta# can be calculated by use of the formula

\[L_\beta =T L_\alpha T^{-1}\]

where #T = {}_{\beta}I_{\alpha}# is the *transition matrix* of the basis #\alpha# to the basis #\beta#. This means we only have to choose the coordinates once, after that we are able to define all other matrix representations of the linear map #L# by only calculating with matrices. We make this explicit in the following theorem.

Basis transitions in terms of matricesLet #\alpha# be a basis of an #n#-dimensional vector space #V#, where #n# is a natural number, and let #L: V\to V# be a linear map with matrix #A# with respect to #\alpha#.

An #(n\times n)#-matrix #B# is the matrix of #L# with respect to a basis #\beta# for #V# if and only if there exists an invertible #(n\times n)#-matrix #T# with

\[B =T A T^{-1}\]

In particular, the determinant of every matrix determining #L# has the same value, so that we can speak of the **determinant** of the linear map #L#.

We give a special name, conjugate, to matrices #A# and #B# who are both from a given linear map. We will also show that this relation satisfies three important properties, summarized in the notion of equivalence relation:

ConjugateLet #n# be a natural number. Two #(n\times n)#-matrices #A# and #B# are said to be **conjugate** if there exists an invertible #(n\times n)#-matrix #T# with #B = T A\, T^{-1}#. We say that the matrix #T# **conjugates** the matrix #A# to #B# and #T# is called the **conjugator**.

Being conjugate is an **equivalence relation**; this means that it has the following three properties for each three #(n\times n)#-matrices #A#, #B#, #C#:

**Reflexivity**: #A# is conjugate with itself (hence, with #A#)**Symmetry**: If #A# and #B# are conjugate, then #B# and #A# are also conjugate**Transitivity**: If #A# and #B# are conjugate, and #B# and #C# are conjugate, then #A# and #C# are also conjugate.

The matrix #T# is invertible since #\det(T)\ne0#. It is easy to verify that #T# satisfies \(B = T\, A\, T^{-1}\):

\[\begin{array}{rcl} T\, A\, T^{-1} &= & \matrix{-3 & -42 \\ 0 & 24 \\ }\,\matrix{1&2\\ 3 &4} \, \matrix{-{{1}\over{3}} & -{{7}\over{12}} \\ 0 & {{1}\over{24}} \\ }\\

&= & \matrix{-129 & -174 \\ 72 & 96 \\ } \, \matrix{-{{1}\over{3}} & -{{7}\over{12}} \\ 0 & {{1}\over{24}} \\ }\\ &=& \matrix{43 & 68 \\ -24 & -38 \\ }\\ &=& B

\end{array}\]

To find #T#, we first solve the matrix equation:

\[\begin{array}{rcl} B &=& T\, A\, T^{-1}\\

&&\phantom{xxx}\color{blue}{\text{the equation that expresses conjugacy}}\\

B \, T&=& T\, A\\

&&\phantom{xxx}\color{blue}{\text{both sides of right hand side multiplied by }T}\\

\matrix{68 z+43 x & 43 y+68 w \\ -38 z-24 x & -24 y-38 w \\ }&=& \matrix{3 y+x & 4 y+2 x \\ z+3 w & 2 z+4 w \\ }\\

&&\phantom{xxx}\color{blue}{\text{matrices multiplied, where }T=\matrix{x&y\\ z&w}}\\

\end{array}\] The solution of this system of linear equations in #x#, #y#, #z#, and #w# is

\[

x={{-13 z-w}\over{8}} ,\quad y={{-z-21 w}\over{12}}

\] Hence, we have the free parameters # w , z #. If we choose # w=24 , z=0 #, then we find # x=-3 , y=-42 #, so

\[ T = \matrix{x&y\\ z&w} = \matrix{-3 & -42 \\ 0 & 24 \\ }\] This answer works because the matrix is invertible.

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