science

Symmetric tensor

Symmetric tensor is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Symmetric tensor rather than just read about it. In short: In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T ( v σ 1 , v σ 2 , … , v σ r ) {\displaystyle T(v_{1},v_{2},\ldots ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\ldots ,v_{\sigma r})} for every permutation σ of the symbols {1, 2, ..., r}. Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with…

Key takeaways

Symmetric tensor belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Symmetric tensor to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Symmetric tensor from memory before moving on to harder problems.

Reference excerpt

In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments:

T ( v 1 , v 2 , … , v r ) = T ( v σ 1 , v σ 2 , … , v σ r ) {\displaystyle T(v_{1},v_{2},\ldots ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\ldots ,v_{\sigma r})}

for every permutation σ of the symbols {1, 2, ..., r}. Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies

T i 1 i 2 ⋯ i r = T i σ 1 i σ 2 ⋯ i σ r . {\displaystyle T_{i_{1}i_{2}\cdots i_{r}}=T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma r}}.}

The space of symmetric tensors of order r on a finite-dimensional vector space V is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics.

Definition Let V be a vector space and

T ∈ V ⊗ k {\displaystyle T\in V^{\otimes k}}

a tensor of order k. Then T is a symmetric tensor if

τ σ T = T {\displaystyle \tau _{\sigma }T=T\,}

for the braiding maps associated to every permutation σ on the symbols {1,2,...,k} (or equivalently for every transposition on these symbols). Given a basis {ei} of V, any symmetric tensor T of rank k can be written as

T = ∑ i 1 , … , i k = 1 N T i 1 i 2 ⋯ i k e i 1 ⊗ e i 2 ⊗ ⋯ ⊗ e i k {\displaystyle T=\sum _{i_{1},\ldots ,i_{k}=1}^{N}T_{i_{1}i_{2}\cdots i_{k}}e^{i_{1}}\otimes e^{i_{2}}\otimes \cdots \otimes e^{i_{k}}}

for some unique list of coefficients T i 1 i 2 ⋯ i k {\displaystyle T_{i_{1}i_{2}\cdots i_{k}}} (the components of the tensor in the basis) that are symmetric on the indices. That is to say

T i σ 1 i σ 2 ⋯ i σ k = T i 1 i 2 ⋯ i k {\displaystyle T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma k}}=T_{i_{1}i_{2}\cdots i_{k}}}

for every permutation σ. The space of all symmetric tensors of order k defined on V is often denoted by Sk(V) or Symk(V). It is itself a vector space, and if V has dimension N then the dimension of Symk(V) is the binomial coefficient

dim ⁡ Sym k ⁡ ( V ) = ( N + k − 1 k ) . {\displaystyle \dim \operatorname {Sym} ^{k}(V)={N+k-1 \choose k}.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Symmetric tensor

Start with the simplest possible case. Write down what Symmetric tensor claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Symmetric tensor before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Symmetric tensor ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Symmetric tensor

In research: Symmetric tensor appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Symmetric tensor in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Symmetric tensor is common in secondary-school and first-year university syllabi. It links to neighbouring topics Tensors, so understanding it makes those chapters shorter.
In everyday life: Look for Symmetric tensor outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Symmetric tensor in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Symmetric tensor means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Symmetric tensor out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Symmetric tensor in simple terms?

In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T ( v σ 1 , v σ 2 , … , v σ r ) {\displaystyle T(v_{1},v_{2},\ldots ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\ldots ,v_{\sigma r})} for every permutation σ o…

Why does Symmetric tensor matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Symmetric tensor?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Symmetric tensor.