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Introduction to Tensor Algebra
Krishna Kannan
Assistant ProfessorDepartment of Mechanical Engineering
IIT Madras
February 28, 2012
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 1 / 13
Outline
1 Algebra of vectors
2 Algebra of Second Order Tensors
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 2 / 13
Outline
1 Algebra of vectors
2 Algebra of Second Order Tensors
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 2 / 13
Algebra of vectors
1 Algebra of vectors
2 Algebra of Second Order Tensors
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 3 / 13
Algebra of vectors
Definition of a vector space
What is a vector ?
A set of objects V = {u, v,w, . . .} endowed with addition+ : V × V → V and scalar multiplication · : R× V → V, where R isthe set of real numbers, is said to form a vector space if it satisfies:
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 4 / 13
Algebra of vectors
Definition of a vector space
What is a vector ?
A set of objects V = {u, v,w, . . .} endowed with addition+ : V × V → V and scalar multiplication · : R× V → V, where R isthe set of real numbers, is said to form a vector space if it satisfies:
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 4 / 13
Algebra of vectors
Definition of a vector space
Addition of vectorsThe sum of two vectors yields a new vector, which has the followingproperties:
u + v = v + u, (1)
(u + v) + w = u + (v + w), (2)
u + o = u, (3)
u + (−u) = o, (4)
where o denotes the zero vector.
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 5 / 13
Algebra of vectors
Definition of a vector space
Scalar MultiplicationThen the scalar multiplication αu produces a new vector with thefollowing properties:
(αβ)u = α(βu), (5)
(α + β)u = αu + βu, (6)
α(u + v) = αu + αv, (7)
where α, β are arbitrary scalars.Any object∈ V that satisfies all the above axioms is called a vector.
Where in the above definition is the idea of magnitude and direction?
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 6 / 13
Algebra of vectors
Definition of an inner product space
An inner product is a scalar valued function f : V × V → R, whichsatisfies:
u · v = v · u, (8)
u · o = 0, (9)
u · (αv + βw) = α(u · v) + β(u ·w), (10)
u · u > 0, ∀u 6= 0, and u · u = 0, if and only if u = 0,(11)
A vector space that is endowed with an inner product is called an innerproduct space.
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 7 / 13
Algebra of vectors
Definition of a normed vector space
A norm is a scalar valued function f : V → R, which satisfies:
‖u‖ ≥ 0, ∀ u ∈ V (12)
‖u‖ = 0 iff u = 0 (13)
‖λu‖ = |λ| ‖u‖ , ∀ u ∈ V and ∀λ ∈ R (14)
‖u + v‖ ≤ ‖u‖+ ‖v‖ , ∀ u ∈ V, ∀ v ∈ V (15)
For example, the quantity |u| = (u · u)1/2 satisfies all the requirements of anorm.Any Innner product space endowed with a norm is called a normed vectorspace.
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 8 / 13
Algebra of Second Order Tensors
1 Algebra of vectors
2 Algebra of Second Order Tensors
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 9 / 13
Algebra of Second Order Tensors
Definition of Second order tensor
Let Lin(V,V) represent the set of all linear transforms (functions) fromthe vector space V to V.One can show that Lin(V,V) is a new vector space by appropriatelydefining addition and scalar multiplication.Objects∈ Lin(V,V) are second order tensors.
A second order tensor A may be thought of as a linear operator thatacts on a vector u generating a vector v
We write v = Au which defines a linear transformation that assigns avector v to each vector u
A is linear we have
A(αu + βv) = αAu + βAv, (16)
for all vectors u, v and all scalars α, β
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 10 / 13
Algebra of Second Order Tensors
Definition of Second order tensor
Let Lin(V,V) represent the set of all linear transforms (functions) fromthe vector space V to V.One can show that Lin(V,V) is a new vector space by appropriatelydefining addition and scalar multiplication.Objects∈ Lin(V,V) are second order tensors.
A second order tensor A may be thought of as a linear operator thatacts on a vector u generating a vector v
We write v = Au which defines a linear transformation that assigns avector v to each vector u
A is linear we have
A(αu + βv) = αAu + βAv, (16)
for all vectors u, v and all scalars α, β
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 10 / 13
Algebra of Second Order Tensors
Definition of Second order tensor
Let Lin(V,V) represent the set of all linear transforms (functions) fromthe vector space V to V.One can show that Lin(V,V) is a new vector space by appropriatelydefining addition and scalar multiplication.Objects∈ Lin(V,V) are second order tensors.
A second order tensor A may be thought of as a linear operator thatacts on a vector u generating a vector v
We write v = Au which defines a linear transformation that assigns avector v to each vector u
A is linear we have
A(αu + βv) = αAu + βAv, (16)
for all vectors u, v and all scalars α, β
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 10 / 13
Algebra of Second Order Tensors
Addition, Subtraction and Scalar Multiplication
If A and B are two second order tensors, we can define the sum A + B,the difference A − B and the scalar multiplication αA by the rules
(A± B)u = Au± Bu, (17)
(αA)u = α(Au), (18)
where u denotes an arbitrary vector
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 11 / 13
Algebra of Second Order Tensors
Identity and Zero Second order tensor
Second order unit (or identity) tensor:
1u = u (19)
Second order zero tensor:
0u = o (20)
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 12 / 13
Algebra of Second Order Tensors
Identity and Zero Second order tensor
Second order unit (or identity) tensor:
1u = u (19)
Second order zero tensor:
0u = o (20)
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 12 / 13
Algebra of Second Order Tensors
Definitions
Positive semi-definite:
v · Av ≥ 0 holds for all nonzero vectors v
Positive Definite:
v · Av > 0 holds for all nonzero vectors v
Negative semi-definite:
v · Av ≤ 0 holds for all nonzero vectors v
Negative Definite:
v · Av < 0 holds for all nonzero vectors v
Define orthogonal tensor, and trace and determinant of a second ordertensor
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 13 / 13
Algebra of Second Order Tensors
Definitions
Positive semi-definite:
v · Av ≥ 0 holds for all nonzero vectors v
Positive Definite:
v · Av > 0 holds for all nonzero vectors v
Negative semi-definite:
v · Av ≤ 0 holds for all nonzero vectors v
Negative Definite:
v · Av < 0 holds for all nonzero vectors v
Define orthogonal tensor, and trace and determinant of a second ordertensor
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 13 / 13
Algebra of Second Order Tensors
Definitions
Positive semi-definite:
v · Av ≥ 0 holds for all nonzero vectors v
Positive Definite:
v · Av > 0 holds for all nonzero vectors v
Negative semi-definite:
v · Av ≤ 0 holds for all nonzero vectors v
Negative Definite:
v · Av < 0 holds for all nonzero vectors v
Define orthogonal tensor, and trace and determinant of a second ordertensor
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 13 / 13
Algebra of Second Order Tensors
Definitions
Positive semi-definite:
v · Av ≥ 0 holds for all nonzero vectors v
Positive Definite:
v · Av > 0 holds for all nonzero vectors v
Negative semi-definite:
v · Av ≤ 0 holds for all nonzero vectors v
Negative Definite:
v · Av < 0 holds for all nonzero vectors v
Define orthogonal tensor, and trace and determinant of a second ordertensor
Krishna Kannan (IIT Madras) Introduction to Tensor Algebra February 28, 2012 13 / 13
