Alqahtani, Refah2021-11-122021-11-122021https://hdl.handle.net/2097/41775The focus of this report is to study fourth-order tensors and generalize some results from the linear algebra of second-order tensors to fourth-order tensors. A fourth-order tensor can be viewed as a linear map from second-order tensors to second-order tensors. We provide an orthonormal basis for the vector space of fourth-order tensors and use it to represent any fourth-order tensor by a fourth-dimensional array, which represents its components' form. An inner product and norm are provided for this vector space. Composition of linear maps gives rise to multiplication of fourth-order tensors, which we present in components' form. We study different kinds of symmetries for fourth-order tensors, in particular, major symmetry and minor symmetry. We provide an isomorphism between the vector space of fourth-order tensors and the vector space of second-order tensors of the same dimension. We use this isomorphism to prove a spectral theorem for fourth-order tensors that possess major symmetry.en-US© the author. This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).http://rightsstatements.org/vocab/InC/1.0/Fourth-order tensorsReport