# Finite Element Procedures – Matrix, Vectors, Tensors

## Special Matrix

symmetric matrix

identity matrix / unite matrix

symmetric banded matrix

the following matrix is a symmetric banded matrix of order 5 and the half-bandwidth is 2.

${\rm{A}} = \left[ {\begin{array}{*{20}{c}} 3&2&1&0&0\\ 2&3&4&1&0\\ 1&4&5&6&1\\ 0&1&6&7&4\\ 0&0&1&4&3 \end{array}} \right]$

diagonal matrix: nonzero elements only on the diagonal of the matrix

upper half of the matrix

inverse matrix, the inverse of a matrix

partitioning of matrix

the trace and determinant of a matrix: only defined if the matrix is square

tr(A) = sum of elements on the diagonal

det A = determinant

det (BC…F) = (det B)(det C)…(det F)

orthogonal matrix:${{\rm{P}}^T} = {{\rm{P}}^{ – 1}}$

## Vectors

change of basis: transformation that corresponds to a change of basis$y = {\rm{A}}x = {{\rm{P}}^{ – 1}}{\rm{AP}}$

rotation matrix (orthogonal): In general case, this rotation is carried out in the n-dimensional space.

${\rm{P}} = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&{ – \sin \theta }\\ {\sin \theta }&{\cos \theta } \end{array}} \right]$

reflection matrix (orthogonal)

${\rm{P}} = {\rm{I}} – \alpha {\rm{v}}{{\rm{v}}^T};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha = \frac{2}{{{{\rm{v}}^T}{\rm{v}}}}$

where v can be arbitrary. This matrix is called a reflection matrix, because the vector Pw is the reflection of vector w in the plane to which v is orthogonal.

## Tensor

Cartesian tensors: tensors represented in rectangular Cartesian coordinate frames.

Scalar: an entity is called a scalar if it has only a single component in the coordinates and this component does not change under a coordinate transformation.

A vector or tensor of first order: if it has three components $\xi _i$ in the unprimed frame and three components $\xi _i^{,}$ in the primed frame, and if these components are related by characteristic law (using the summation convention).

$\xi _i^{,} = {p_{ik}}{\xi _k}$

the scalar product of the vectors u and v

${\rm{u}} \times {\rm{v}} = \left| {\rm{u}} \right|\left| {\rm{v}} \right|\cos \theta = {u_i}{v_i}$

the dot product of the vectors u and v

${\rm{w}} = \det \left[ {\begin{array}{*{20}{c}} {{{\rm{e}}_1}}&{{{\rm{e}}_2}}&{{{\rm{e}}_3}}\\ {{u_1}}&{{u_2}}&{{u_3}}\\ {{v_1}}&{{v_2}}&{{v_3}} \end{array}} \right]$

the scalar and dot product of vectors are frequently employed in finite element analysis to evaluate angles between two given directions and to establish the direction perpendicular to a given plane.

A second order tensor: it has nine components $t_{ij}$, i = 1, 2, 3 and j = 1, 2, 3 in the unprimed frameand nine components $t_{ij}^{,}$ in the primed frame and if these components are related by the characteristic law

$t_{ij}^{,} = {p_{ij}}{p_{jk}}{t_{kl}}$