Let A be a symmetric tensor with Λi, as eigenvalues and v,· as corresponding eigenvectors. Show that A can be represented as

Question

Let A be a symmetric tensor with [latex]\Lambda _i[/latex], as eigenvalues and [latex]v_{i}[/latex]· as corresponding eigenvectors. Show that A can be represented as

[latex] extbf{A}=\sum\limits_{k=1}^{3}\ \bf{}\Lambda_{k}(v_{k} \otimes v_{k})[/latex] (2.13.23)

This is known as the spectral representation of A.

Accepted Answer

In Example 2.7.3, it has been shown that every tensor A has the following representation in a coordinate system having [latex]e_i[/latex], as base vectors: [latex] extbf{A}=a_{ij}e_{i}\otimes e_{j}[/latex]. Hence, with respect to a coordinate system with the eigenvectors v, as base vectors, a symmetric tensor A has the representation

[latex] extbf{A}=a_{ij} extbf{v}_{i}\otimes extbf{v}_{j}[/latex] (2.13.24)

Also, with respect to this coordinate system, the matrix of A is given by
(2.3.19). Using (2.3.19), the representation (2.13.24) becomes

[latex]a^{,}_{i}b^{,}_{i}=(\alpha_{ip}a_{p})(\alpha_{iq}b_{q})=(\alpha_{ip}\alpha_{iq})a_{p}b_{q}=\delta_{pq}a_{p}b_{p}=a_{i}b_{i}[/latex] (2.3.19)

[latex] extbf{A}=\Lambda_{1}(v_{1} \otimes v_{1})+\Lambda_{2}(v_{2} \otimes v_{2})+\Lambda_{3}(v_{3} \otimes v_{3})[/latex]

which is (2.13.23).

Question 2.13.5: Let A be a symmetric tensor with Λi, as eigenvalues and v,· ...

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