Common Operations

The most common operations between scalars, vectors, and 2nd-order tensors (from now on, simply referred to as tensors) are presented below in matrix notation. All components are defined with respect to some basis vectors ${e_{1}, e_{2}, e_{3}}$ in the 3D Euclidean space.

Vectors

Multiplication by a Scalar: The product of a vector $a$ with a scalar $α$ defines a vector:; Figure 1. EQUATION_DISPLAY

$α a = α (\begin{matrix} {\begin{matrix} a \end{matrix}}_{1} \\ {\begin{matrix} a \end{matrix}}_{2} \\ {\begin{matrix} a \end{matrix}}_{3} \end{matrix}) = (\begin{matrix} {\begin{matrix} α a \end{matrix}}_{1} \\ {\begin{matrix} α a \end{matrix}}_{2} \\ {\begin{matrix} α a \end{matrix}}_{3} \end{matrix})$
(5190)
Sum of two Vectors: The sum of two vectors $a$ and $b$ defines a vector:; Figure 2. EQUATION_DISPLAY

$a + b = (\begin{matrix} {\begin{matrix} a \end{matrix}}_{1} \\ {\begin{matrix} a \end{matrix}}_{2} \\ {\begin{matrix} a \end{matrix}}_{3} \end{matrix}) + (\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}) = (\begin{matrix} {\begin{matrix} a \end{matrix}}_{1} + b_{1} \\ {\begin{matrix} a \end{matrix}}_{2} + b_{2} \\ {\begin{matrix} a \end{matrix}}_{3} + b_{3} \end{matrix})$
(5191)
Scalar Product of two Vectors: The scalar product between two vectors $a$ and $b$ is the matrix product between the row vector representation of $a$ and the column vector representation of $b$ , which defines a scalar:; Figure 3. EQUATION_DISPLAY

$a \cdot b = (\begin{matrix} {\begin{matrix} a \end{matrix}}_{1} & {\begin{matrix} a \end{matrix}}_{2} & {\begin{matrix} a \end{matrix}}_{3} \end{matrix}) (\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}) = \begin{matrix} {\begin{matrix} a \end{matrix}}_{1} \end{matrix} \begin{matrix} b_{1} \end{matrix} + \begin{matrix} {\begin{matrix} a \end{matrix}}_{2} \end{matrix} \begin{matrix} b_{2} \end{matrix} + \begin{matrix} {\begin{matrix} a \end{matrix}}_{3} \end{matrix} \begin{matrix} b_{3} \end{matrix}$
(5192); As vectors are usually defined in column representation, the scalar product is often written as $a \cdot b = a^{T} b = b^{T} a$ , where $T$ indicates the matrix transpose. The transpose of a column vector is a row vector, and the transpose of a row vector is a column vector.; The magnitude of a vector $a$ can be written as:; Figure 4. EQUATION_DISPLAY

$| a | = \sqrt{a \cdot a}$
(5193)
Vector Product of two Vectors: The vector product between two vectors $a$ and $b$ defines a vector:; Figure 5. EQUATION_DISPLAY

$a \times b = (\begin{matrix} {\begin{matrix} a \end{matrix}}_{2} b_{3} - {\begin{matrix} a \end{matrix}}_{3} b_{2} \\ {\begin{matrix} a \end{matrix}}_{3} b_{1} - {\begin{matrix} a \end{matrix}}_{1} b_{3} \\ {\begin{matrix} a \end{matrix}}_{1} b_{2} - {\begin{matrix} a \end{matrix}}_{2} b_{1} \end{matrix})$
(5194); which is obtained from the formal determinant:; Figure 6. EQUATION_DISPLAY

$\det (\begin{matrix} {\begin{matrix} e \end{matrix}}_{1} & {\begin{matrix} e \end{matrix}}_{2} & {\begin{matrix} e \end{matrix}}_{3} \\ {\begin{matrix} a \end{matrix}}_{1} & {\begin{matrix} a \end{matrix}}_{2} & {\begin{matrix} a \end{matrix}}_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix})$
(5195)
Dyadic Product of two Vectors: The dyadic product, or tensor product, between two vectors $a$ and $b$ defines a tensor:; Figure 7. EQUATION_DISPLAY

$a b \equiv a \otimes b = (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}) (\begin{matrix} b_{1} & b_{2} & b_{3} \end{matrix}) = (\begin{matrix} a_{1} b_{1} & a_{1} b_{2} & a_{1} b_{3} \\ a_{2} b_{1} & a_{2} b_{2} & a_{2} b_{3} \\ a_{3} b_{1} & a_{3} b_{2} & a_{3} b_{3} \end{matrix})$
(5196); The notation $a b$ assumes that $a$ is represented by a column vector and $b$ is represented by a row vector. The tensor product is often written as $a b^{T}$ , which assumes that both $a$ and $b$ are represented by column vectors.
Gradient of a Scalar: The gradient of a scalar $φ$ defines a vector:; Figure 8. EQUATION_DISPLAY

$\nabla φ = (\begin{matrix} \frac{\partial φ}{\partial x} \\ \frac{\partial φ}{\partial y} \\ \frac{\partial φ}{\partial z} \end{matrix})$
(5197)
Gradient of a Vector: The gradient of a vector $a$ defines a tensor:; Figure 9. EQUATION_DISPLAY

$\nabla a = (\begin{matrix} \frac{\partial a_{1}}{\partial x} & \frac{\partial a_{1}}{\partial y} & \frac{\partial a_{1}}{\partial z} \\ \frac{\partial a_{2}}{\partial x} & \frac{\partial a_{2}}{\partial y} & \frac{\partial a_{2}}{\partial z} \\ \frac{\partial a_{3}}{\partial x} & \frac{\partial a_{3}}{\partial y} & \frac{\partial a_{3}}{\partial z} \end{matrix})$
(5198)
Divergence of a Vector: The divergence of a vector $a$ defines a scalar:; Figure 10. EQUATION_DISPLAY

$\nabla\cdot a = \begin{matrix} \frac{\partial a_{1}}{\partial x} \end{matrix} + \begin{matrix} \frac{\partial a_{2}}{\partial y} \end{matrix} + \begin{matrix} \frac{\partial a_{3}}{\partial z} \end{matrix}$
(5199)
Curl of a Vector: The curl of a vector $a$ defines a vector:; Figure 11. EQUATION_DISPLAY

$\nabla\times a = (\begin{matrix} \frac{\partial {\begin{matrix} a \end{matrix}}_{3}}{\partial y} - \frac{\partial {\begin{matrix} a \end{matrix}}_{2}}{\partial z} \\ \frac{\partial {\begin{matrix} a \end{matrix}}_{1}}{\partial z} - \frac{\partial {\begin{matrix} a \end{matrix}}_{3}}{\partial x} \\ \frac{\partial {\begin{matrix} a \end{matrix}}_{2}}{\partial x} - \frac{\partial {\begin{matrix} a \end{matrix}}_{1}}{\partial y} \end{matrix})$
(5200); which is obtained from the formal determinant:; Figure 12. EQUATION_DISPLAY

$\det (\begin{matrix} {\begin{matrix} e \end{matrix}}_{1} & {\begin{matrix} e \end{matrix}}_{2} & {\begin{matrix} e \end{matrix}}_{3} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ a_{1} & a_{2} & a_{3} \end{matrix})$
(5201)

Tensors

Multiplication by a Scalar

The product of a tensor

A

with a scalar

α

defines a tensor:

Figure 13. EQUATION_DISPLAY

α A = α (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}) = (\begin{matrix} {αA}_{11} & {αA}_{12} & {αA}_{13} \\ {αA}_{21} & {αA}_{22} & {αA}_{23} \\ {αA}_{31} & {αA}_{32} & {αA}_{33} \end{matrix})

(5202)

Sum of two Tensors

The sum of two tensors

A

and

B

defines a tensor:

Figure 14. EQUATION_DISPLAY

\begin{matrix} A + B & = (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}) + (\begin{matrix} B_{11} & B_{12} & B_{13} \\ B_{21} & B_{22} & B_{23} \\ B_{31} & B_{32} & B_{33} \end{matrix}) = \\ = (\begin{matrix} A_{11} + B_{11} & A_{12} + B_{12} & A_{13} + B_{13} \\ A_{21} + B_{21} & A_{22} + B_{22} & A_{23} + B_{23} \\ A_{31} + B_{31} & A_{32} + B_{32} & A_{33} + B_{33} \end{matrix}) \end{matrix}

(5203)

Product of a Tensor with a Vector

The product of a tensor

A

with a vector

b

defines a vector:

Figure 15. EQUATION_DISPLAY

A b \equiv A \cdot b = (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}) (\begin{matrix} {\begin{matrix} b \end{matrix}}_{1} \\ {\begin{matrix} b \end{matrix}}_{2} \\ {\begin{matrix} b \end{matrix}}_{3} \end{matrix}) = (\begin{matrix} A_{11} {\begin{matrix} b \end{matrix}}_{1} + A_{12} {\begin{matrix} b \end{matrix}}_{2} + A_{13} {\begin{matrix} b \end{matrix}}_{3} \\ {\begin{matrix} A_{21} {\begin{matrix} b \end{matrix}}_{1} + A_{22} {\begin{matrix} b \end{matrix}}_{2} + A_{23} b \end{matrix}}_{3} \\ A_{31} {\begin{matrix} b \end{matrix}}_{1} + A_{32} {\begin{matrix} b \end{matrix}}_{2} + A_{33} {\begin{matrix} b \end{matrix}}_{3} \end{matrix})

(5204)

Therefore, tensors define linear transformations between vectors.

Product of two Tensors

The product of two tensors

A

and

B

defines a tensor:

Figure 16. EQUATION_DISPLAY

\begin{matrix} A B & \equiv A \cdot B = (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}) (\begin{matrix} B_{11} & B_{12} & B_{13} \\ B_{21} & B_{22} & B_{23} \\ B_{31} & B_{32} & B_{33} \end{matrix}) = \\ = (\begin{matrix} A_{11} B_{11} + A_{12} B_{21} + A_{13} B_{31} & A_{11} B_{12} + A_{12} B_{22} + A_{13} B_{32} & A_{11} B_{13} + A_{12} B_{23} + A_{13} B_{33} \\ A_{21} B_{11} + A_{22} B_{21} + A_{23} B_{31} & A_{21} B_{12} + A_{22} B_{22} + A_{23} B_{32} & A_{21} B_{13} + A_{22} B_{23} + A_{23} B_{33} \\ A_{31} B_{11} + A_{32} B_{21} + A_{33} B_{31} & A_{31} B_{12} + A_{32} B_{22} + A_{33} B_{32} & A_{31} B_{13} + A_{32} B_{23} + A_{33} B_{33} \end{matrix}) \end{matrix}

(5205)

Therefore, tensors also define linear transformations between tensors.

Inner Product of two Tensors

The inner product, or double dot product, of two tensors

A

and

B

defines a scalar:

Figure 17. EQUATION_DISPLAY

A : B = A_{11} B_{11} + A_{12} B_{12} + A_{13} B_{13} + ...

(5206)

or, in index notation:

Figure 18. EQUATION_DISPLAY

A : B = A_{ij} B_{ij}

(5207)

The inner product is a tensor invariant, as its value remain unchanged under a change of basis.

Divergence of a Tensor

The divergence of a tensor

A

defines the vector:

Figure 19. EQUATION_DISPLAY

\begin{matrix} \nabla\cdot A = (\begin{matrix} \frac{\partial A_{11}}{\partial x} + \frac{\partial A_{21}}{\partial y} + \frac{\partial A_{31}}{\partial z} \\ \frac{\partial A_{12}}{\partial x} + \frac{\partial A_{22}}{\partial y} + \frac{\partial A_{32}}{\partial z} \\ \frac{\partial A_{13}}{\partial x} + \frac{\partial A_{23}}{\partial y} + \frac{\partial A_{33}}{\partial z} \end{matrix})^{T} \end{matrix}

(5208)

Eigenvalues and Eigenvectors of a Tensor

Given a tensor

A

, the scalars

λ_{i}

and vectors

v_{i}

that satisfy the relation:

Figure 20. EQUATION_DISPLAY

(A - λ_{i} I) v_{i} = 0

(5209)

are called the eigenvalues and eigenvectors of

A

, respectively. The three eigenvalues are determined by solving the characteristic equation:

Figure 21. EQUATION_DISPLAY

\det (A - λ_{i} I) = 0

(5210)

Inverse of a Tensor

Eqn. (5204) and Eqn. (5205) define tensors as linear transformations between vectors, or tensors. Given a tensor

A

, the inverse transformation

A^{- 1}

is defined as:

Figure 22. EQUATION_DISPLAY

A^{- 1} A = I

(5211)

where

I

is the identity tensor:

Figure 23. EQUATION_DISPLAY

I = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})

(5212)

The inverse tensor,

A^{- 1}

, exists if and only if

\det (A) \neq 0

, where:

Figure 24. EQUATION_DISPLAY

\begin{matrix} \det (A) & = \det (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}) = \\ \begin{matrix} {= A}_{11} \end{matrix} (\begin{matrix} A_{22} A_{33} - A_{23} A_{32} \end{matrix}) + A_{12} (\begin{matrix} A_{23} A_{31} - A_{21} A_{33} \end{matrix}) + A_{13} (\begin{matrix} A_{21} A_{32} - A_{31} A_{22} \end{matrix}) \end{matrix}

(5213)

Symmetry

A tensor

A

is called a symmetric tensor when

A = A^{T}

, that is:

Figure 25. EQUATION_DISPLAY

(\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}) = (\begin{matrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{matrix})

(5214)

Tensor Invariants

Given a tensor

A

, it is possible to define three independent invariants, that is, scalar quantities that remain unchanged under a change of coordinate system:

Figure 26. EQUATION_DISPLAY

I_{0} = t r (A) = A_{11} + A_{22} + A_{33} = A_{i i}

(5215)

Figure 27. EQUATION_DISPLAY

I_{1} = [tr {(A)}^{2} - tr (A^{2})] / 2

(5216)

Figure 28. EQUATION_DISPLAY

I_{2} = \det (A)

(5217)

The invariant

I_{0}

is called the Trace of the tensor

A

Tensor Norms

Given a tensor

A

, of dimension

m \times n

, it is possible to define its norm in various ways:

Infinity Norm: Maximum absolute row sum of $A$ :
Figure 29. EQUATION_DISPLAY

$‖ A ‖_{\infty} = \max_{1 \leq j \leq m} \sum_{i = 1}^{n} | A_{i j} |$
(5218)
1-Norm: Maximum absolute column sum of $A$ :
Figure 30. EQUATION_DISPLAY

$‖ A ‖_{1} = \max_{1 \leq j \leq n} \sum_{i = 1}^{m} | A_{i j} |$
(5219)
2-Norm: Figure 31. EQUATION_DISPLAY

$‖ A ‖_{2} = \max_{o \leq i \leq 3} | λ_{i} |$
(5220)
Frobenius Norm: Figure 32. EQUATION_DISPLAY

$‖ A ‖_{F} = \sqrt{t r (A^{2})}$
(5221)