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TENSOR DIAGONALIZATION 19
becomes
u. 6; = Al@{. (4.57)
The A1 factor is called an eigenvalue of ?F. An eigenvalue results when an operation
on an object produces a constant,the eigenvalue, times the original object. The primed basis vector is called an eigenvector. Now, we introduce the special unit tensor I, which is defined as
- 1 = a,.@.@. ‘I 1 J (4.58)
so that
**1. V = v.** (4.59)
Represented as a matrix, T is simply
1 + [ 1 ] = 0 1 0. (4.60)
[: :]
Using the 7 tensor, Equation 4.57 can be rearranged to read
(2 - A l i ) * 6; = 0. (4.61)
Expressing in the unprimed system, Equation 4.61 can be written^ in^ subscript notation as
Equation 4.29 and some rearrangement yields
@ i ( c r , j - A 1 G i j ) a l j = 0, (4.63)
where the al are three of the unknown elements of the transformation matrix relating the original coordinate systems to the one in which The LHS of Equation 4.63 is a vector, and for it to be zero, each of its components must be zero. Each component involves a sum over the index j. Equation 4. therefore becomes three equations which can be written in matrix array notation as
is diagonal.
u 1 3
~ 2 - 2 A I c 2 3 ] [ I! : ] = [a]. (4.64)
g 3 2 u 3 3 -A
In order for a set of linear, homogeneous equations such as those in Equation 4. to have a solution, the determinant of the coefficients must be zero:
80 INTRODUCTION TO TENSORS
g 1 1 - A1 u 1 2
a 2 1 ~ 2 - 2 A1 u13 g 2 3 I = 0. (4.65) 1 a 3 1 a 3 2 u 3 3 - A1 det
This results in a third-order equation for A1 which will generate three eigenvalues. Select one of these values, it does not matter which one because the other two will be used later, and call it A]. Inserting this value into Equation 4.64 allows a solution for a l l , a12, and a 1 3 to within an arbitrary constant. These are three of the elements of the transformation matrix between the primed and unprimed systems that we seek. These three elements also allow the determination of the 6; basis vector to within an arbitrary constant:
e; = alje,. A (4.66)
Requiring @[to be a unit vector determines the arbitrary constant associated with a l l ,
a 1 2 and ~ 1 3 :
( a d 2+ (a12I2 + (d2= 1. (4.67)
Except for an overall arbitrary sign and the degenerate situation discussed below, we have now uniquely determined 6;. The other primed basis vectors and elements of the transformation matrix are obtained in a similar way. The second primed basis vector is determined by forming the dot product in Equation 4.56 using 6;. A matrix equation equivalent to 4. is written with A2, a21, az2, and ~ 2 3 .The resulting determinant equation for A2 is identical to the one for A l , in Equation 4.65. One of the two remaining eigenvalues of this equation is selected for A2 and used to determine a21, a 2 2 , a23, and $4. In a similar way, the last eigenvalue of Equation 4.65 is used for A3 to determine ~ 3 1 , ~ 3 2 ,
is diagonal, is defined by the basis vectors 6 ; , 64, and $:. The elements of ??in this primed system are just the eigenvalues determined from Equation 4.65,
~ 3 3 ,and $4. The primed coordinate system, in which
0 0 A
The matrices of interest in physics and engineering are typically Hermitian. If we allow the possibility for complex matrix elements, a matrix is Hermitian if it is
equal to its complex conjugate transpose. That is, %j = eF. There are two important
properties of Hermitian matrices. First, their eigenvalues are always pure real num- bers. Second, their eigenvectors are always orthogonal. Proofs of these statements are left as exercises at the end of this chapter. The only complication that can arise in the above diagonalization process is a degenerate situation that occurs when two or more of the eigenvalues are identical. Consider the case when Al # A2 = A3. The unique eigenvalue Al determines all,
