## 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

\[t_{ij}^{,} = {p_{ij}}{p_{jk}}{t_{kl}}\]

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