Elements of Tensor Algebra

LetE3be a Euclidean vector space. The mapping

TWE3!E3 (1.39)

islinearif, using the notationT.u/DTu, we have

T.auCbv/DaTuCbTv; (1.40)

for alla; b2 <and for allu;v2E3.

We will let Lin.E3/ denote the set of linear mappings of E3 into E3. These applications are also called endomorphisms, Euclidean double tensors, or 2-tensorsofE3.

TheproductSTof two tensorsSandTis the composition ofSandT; that is, for allu2E3we have

.ST/uDS.Tu/: (1.41)

Generally,ST¤TS; ifST D TS, then the two tensors are said tocommute. The transposeT^T ofTis the new tensor defined by the condition

uT^TvDv.Tu/; 8u;v2E3: (1.42)

1.7 Elements of Tensor Algebra 15 The following properties can be easily established:

.TCS/^T DT^T CS^T; .TS/^T DS^TT^T; .T^T/^T DT:

A tensorTissymmetricif

TDT^T; (1.43)

andskew-symmetricif

TD T^T: (1.44)

Thetensor productof vectorsuandvis the tensoru˝vsuch that

.u˝v/wDu vw; 8w2E3: (1.45)

The set Lin.E3/of tensors defined onE3is itself a vector space with respect to the ordinary operations of addition and the product of a mapping times a real number.

For this space the following theorem holds:

Theorem 1.3. The vector space Lin.E3/ of Euclidean double tensors is 9-dimensional. The tensor systems.e_i˝ej/; .e_i˝e^j/; .eⁱ˝e^j/are bases of Lin.E3/ so that for allu 2 E3, the following representations of Euclidean double tensors hold:

TDT^ijei˝ej DT_jⁱei˝e^j DT_ijeⁱ˝e^j: (1.46) The matrices .T^ij/, .T_jⁱ/, and .T_ij/ are called the contravariant, mixed, and covariant components ofT, respectively. Finally, the relations among the different components are given by the equations

T_ij Dg_{j h}T_i^hDg_ihg_{j k}T^hk: (1.47) Proof. To verify that the system.e_i ˝ej/is a basis of Lin.E3/, it is sufficient to prove that it is a linearly independent system and that any tensorTcan be expressed as a linear combination of the elements of.e_i˝ej/. First, from

^ijei˝ej D0;

and from (1.45), (1.12), we derive

^ijei˝ej.e_h/D^ijei ejehD^ijg_{j h}ei D0

for anyh D 1; 2; 3. But the vectors.e_i/form a basis ofE3, so that the previous relation implies, for any indexi, the homogeneous system

^ijg_{j h}D0

of three equations in the three unknowns^ij, whereiis fixed. Since the determinant det.g_ij/of this system is different from zero, see (1.14), all the unknowns^ij with fixedivanish. From the arbitrariness ofi, the theorem follows.

Similarly, from the linear combination ⁱ_je_i˝e^j D0;

when (1.19) is taken into account, we have

ⁱ_jei ˝e^j.eh/Dⁱ_jei e^j ehDⁱ_jı^j_hei Dⁱ_hei D0;

so thatⁱ_h D 0, for any choice ofi andh. In the same way, the independence of tensors.eⁱ˝e^j/can be proved.

To show that any tensorTcan be written as a linear combination of any one of these systems, we start by noting that, from (1.21) and the linearity ofT, we have

TuDujTe^j:

On the other hand,Te^j is a vector of E3 and therefore can be represented in the following form, see (1.21):

Te^j DT^ijei:

If the definition of covariant components (1.23) is recalled together with (1.45), we have

TuDT^ijujei DT^ijei˝ej.u/:

Finally, owing to the arbitrariness ofu, we obtain TDT^ijei˝ej: To show (1.46)2, it is sufficient to remark that

TuDu^jTej; so that, introducing the notationTej DT_jⁱei, we derive

TuDT_jⁱu^jei:

1.7 Elements of Tensor Algebra 17 Noting that (1.23) and (1.22) hold, we haveu^j D g^{j i}ui Dg^{j i}ei u De^j u, and the previous relation gives

TuDT_jⁱei˝e^j.u/:

From this expression, due to the arbitrariness ofu, (1.46)2 is derived. In a similar way, (1.46)3can be proved.

Relation (1.47) is easily verified using (1.46) and the definition of the reciprocal

basis (1.18), so that the theorem is proved. ut

Example 1.1. Verify that the matrices of contravariant or covariant components of a symmetric (skew-symmetric) tensor are symmetric (skew-symmetric), using the definition of transpose and symmetry.

The symmetry ofT(TDT^T) and (1.42) imply that vTuDuTv;

so that

viT^ijuj DuiT^ijvj DujT^{j i}vi:

The arbitrariness ofuandvleads toT^ij DT^{j i}. In a similar way, it can be proved thatT_ij D T_{j i} whenTis skew-symmetric.

Example 1.2. Using the different representations (1.46), verify that the contravari-ant, mixed and covariant components of the unit or identity tensorI

IuDu; 8u2E3; are given by the following matrices, respectively:

.g^ij/; .ı_jⁱ/; .g_ij/:

Starting from the condition definingI, written in terms of components, .I^ijei ˝ej/.u_he^h/Duieⁱ;

we derive the equation

I^ihuhei Dg^hiuhei;

which implies thatI^ihDg^ih. In a similar way the others results follow.

TheimageofT, denoted by Im.T/, is the subset ofE3such that

Im.T/D fv2E3 j 9u2E3WvDTug; (1.48)

whereas thekernelofT, denoted by Ker.T/, is the subset ofE3such that

Ker.T/D fu2E3jTuD0g: (1.49)

Theorem 1.4. Im.T/and Ker.T/are vector subspaces ofE3. Moreover,Thas an inverse if and only if Ker.T/D f0g:

Proof. In fact, ifv₁ DTu₁2Im.T/andv₂DTu₂2Im.T/, we have av1Cbv2DaTu1CbTu2DT.au1Cbu2/;

so thatav1Cbv22Im.T/:Moreover, ifu1;u22Ker.T/, we haveTu1DTu2D0 and therefore

0DaTu1CbTu2DT.au1Cbu2/;

so thatau1Cbu2 2Ker.T/:Finally, ifThas an inverse, the conditionT.0/ D 0 implies that the inverse image of the zero vector ofE3contains only the zero vector, i.e., Ker.T/D f0g. Conversely, if this condition implied the existence of two vectors u⁰;u⁰⁰2E3such thatTu⁰DTu⁰⁰, we would haveT.u⁰u⁰⁰/D0. From this relation it would follow thatu⁰u⁰⁰2Ker.T/and thereforeu⁰u⁰⁰D0: ut As a consequence of the previous results, we can say thatTis an isomorphism if and only if

Im.T/DE3; Ker.T/D f0g: (1.50)

Theorem 1.5. Let us consider a tensorT2Lin.E3/and a basis.ei/ofE3. Then the following conditions are equivalent:

1. Tis an isomorphism;

2. the vectorsT.ei/represent a basisE3;

3. the representative matrix.T_jⁱ/ofTwith respect to the basis.ei/is not singular.

Proof. 1.H)2. In fact, from the conditionsⁱTei DT.ⁱei/D0and (1.50)2, it follows thatⁱei D0; consequently,ⁱ D0, since.ei/is a basis and the vectors Tei are independent. Moreover, ifTis an isomorphism, then for any v 2 E3

there exists a uniqueu 2 E3 such thatv D TuD uⁱTei, so that the vectorsTei

represent a basis ofE3.

2.H)3. In fact, the conditionⁱTei D 0can be writtenⁱT_i^heh D 0 so that, due to (2), the homogeneous systemⁱT_i^h D 0must admit only the vanishing solution and therefore the matrix.T_jⁱ/is not singular.

3.H)1. In terms of components,vDTuis expressed by the system

vⁱ DT_hⁱu^h: (1.51)

1.7 Elements of Tensor Algebra 19 For any choice of the vectorv 2 E3, this system can be considered as a linear system of n equations with n unknowns u^h, which admits one and only one solution, when.T_jⁱ/is not singular. Consequently,Tis an isomorphism. ut To conclude this section, the transformation rules under a base change.ei/! .e⁰_i/of the components of a Euclidean second-order tensor will be derived. First, from (1.48), we have

TDT^ijei˝ej DT^0hke⁰_h˝e⁰_k; so that, recalling (1.25), we find

.A¹/^h_i.A¹/^k_jT^ije⁰_h˝e⁰_kDT^0hke⁰_h˝e⁰_k: Since the tensors.e⁰_h˝e⁰_k/form a basis of Lin.E3/;we have

T^0hkD.A¹/^h_i.A¹/^k_jT^ij: (1.52) Similarly, starting from (1.46)2;3and taking into account (1.25), we can derive the following transformation formulae for mixed and covariant components:

T_k^0hD.A¹/^h_iA^j_kT_jⁱ; (1.53) T_hk⁰ DAⁱ_hA^j_kTij: (1.54) Remark 1.1. In the following sections we will often use the notationuT. It denotes the linear mappingE3 !E3that, in terms of components, is written

vⁱ Du^jT_jⁱ: (1.55)

When the tensorTis symmetric, we have uTDTu:

Remark 1.2. It is worthwhile noting that the relations (1.51) to (1.54) can be written adopting matrix notation. For instance, in agreement with the convention that the product of two matrices is calculated by rows times columns, the following form can be given to (1.54):

T⁰DA^TTA; (1.56)

where T andT⁰ are the matrices formed with the components ofT andT⁰ with respect to the bases.e_i/and.e⁰_i/, respectively, andAdenotes the matrix of the base changeei !e⁰_i, see (1.25). It is also important to note that in the literature, usually,

the same symbol denotes both the tensorTand its representative matrixT in a fixed basis. Consequently, relation (1.56) is also written as

T⁰DA^TTA:

1.8 Eigenvalues and Eigenvectors of a Euclidean