# Diagonalization

As matrices go, diagonal matrices are easy to work with. Sometimes this ease can be exploited with matrices that are not themselves diagonal.

Let ` A` be a square,

`matrix. If`

*n*×*n*`is a scalar, then the corresponding`

*x***eigenspace**of

`is the nullspace of the`

*A*`matrix`

*n*×*n*`. Evidently the eigenspace is the trivial vector space unless`

*x**I*-*A*`det(`.

*x**I*-*A*) = 0The polynomial

`det( xI - A)`

(which is of degree ` n` in the variable

`) is the`

*x***characteristic polynomial**of

`; its zeros are the`

*A*`eigenvalues`of

`. An`

*A***eigenvector**is a nonzero member of an eigenspace.

The square matrix ` A` is

**diagonalizable**if

*A**P* = *P**D*

for some *invertible* matrix ` P`,
where

`is a diagonal matrix; note then`

*D*`.`

*A*=*P**D**P*^{-1}

**Theorem.**Let

`be an`

*A*`matrix. The following are equivalent:`

*n*×*n*-
has*A*linearly independent eigenvectors.*n* -
is diagonalizable.*A*

To prove this theorem, suppose ` P` is the
matrix

`( p_{1} p_{2}
... p_{n}), `

where each column ` p_{i}` is
in

`. Then`

**R**^{n}
*A**P* = (*A***p**_{1} *A***p**_{2}
... *A***p**_{n})

If ` D` is a diagonal matrix, with diagonal
entries

`, then`

*x*_{i}
*D* =
(*x*_{1}**e**_{1}
*x*_{2}**e**_{2}
... *x*_{n}**e**_{n}) =
(*x*_{1}**e**_{1}
*x*_{2}**e**_{2}
...
*x*_{n}**e**_{n})^{T},

and therefore

*P**D* =
(*x*_{1}**p**_{1}
*x*_{2}**p**_{2}
... *x*_{n}**p**_{n}).

Therefore, ` AP = PD` if
and only if

`for each`

*A***p**_{i}=*x*_{i}**p**_{i}`from`

*i*`1`to

`inclusive.`

*n*## Application to differential equations

As noted earlier, a linear
polynomial is the result of applying to variables the operations
of addition and scalar multiplication; these operations obey the same
algebraic rules as they do in vector
spaces. In a linear **differential** polynomial, the
operation of **differentiation** may be applied as well:
from a polynomial ` f`, the operation produces a
polynomial

`(read `eff-prime'), and it satisfies the following algebraic rules:`

*f**'*-
`(`when*a**f*)*'*=*a**f**'*is a scalar;*a* -
`(`.*f*+*g*)*'*=*f**'*+*g**'*

`is equivalent to the homogeneous linear differential equation`

*y**'*=*a**y*
*y**'* - *a**y* = 0 ;

from calculus its solution is known to be ` y =
ce^{ax}` (also written

`). Here`

*y*=*c*exp(*a**x*)`is the value of`

*c*`when`

*y*`; so we can write the solution thus:`

*x*= 0
*y* =
*y*(0)e^{ax} .

Therefore the system of equations

*y*_{1}*'* - *a*_{1}*y*_{1} = 0,
*y*_{2}*'* - *a*_{2}*y*_{2} = 0, ...,
*y*_{n}*'* - *a*_{n}*y*_{n} = 0

has the solution

*y*_{1} = *y*_{1}(0)e^{(a1x)},
*y*_{2} = *y*_{2}(0)e^{(a2x)}, ...,
*y*_{n} = *y*_{n}(0)e^{(anx)}
.

We can write the last conclusion in matrix form:

`has the solution`

**y***'*-*D***y**=**0**`, where`

**y**= e^{(xD)}**y**(0)-
is the vector**y**`(`of variables;*y*_{1}*y*_{1}...*y*_{n})^{T} is the diagonal matrix whose diagonal entries are*D*,*a*_{1}, ...,*a*_{2}respectively;*a*_{n}-
`e`is the diagonal matrix whose diagonal entries are^{(xD)}`e`,^{(a1x)}`e`, ...,^{(a2x)}`e`respectively.^{(anx)}

`, where`

**y***'*-*A***y**=**0**`is an arbitrary`

*A*`matrix. If this matrix is diagonalizable, say`

*n*×*n*`, then the system has the solution`

*A*=*P**D**P*^{-1}
**y** =
*P*e^{(xD)}*P*^{-1}**y**(0)
.

If one is required to solve such a system, given numerical values for
the entries of ` A`, then one need not
actually calculate the inverse of

`unless the`

*P***initial conditions**

`are specified; even then, to find the vector`

**y**(0)`, one need only solve for`

*P*^{-1}**y**(0)`the equation`

**c**
*P***c** = **y**(0) ,

and then the solution of the original system is ` y =
Pe^{(xD)}c`. (Even if the matrix

`is not diagonalizable, the original system is still soluble, but by a more complicated procedure.)`

*A*