Pages Symmetric Matrix Attributions

Attributions & Verification

Track contributions and verify content correctness

Content

A symmetric matrix is the matrix form of a linear transformation whose action is balanced with respect to the Euclidean [inner product](/page/Inner%20Product).

text admin

When a matrix $A$ is used in expressions such as $\langle Ax,y\rangle$, $x^\top Ay$, or $x^\top Ax$, symmetry determines whether the two vector inputs can be exchanged without changing the scalar measurement.

text admin

This is why symmetric matrices appear across linear algebra, quadratic forms, [inner product spaces](/page/Inner%20Product%20Space), multivariable calculus, probability, and optimization.

text admin

A general real square matrix may rotate, shear, and stretch space in coupled ways.

text admin

A real symmetric matrix has a much more rigid geometry: it admits perpendicular eigenvector directions and real eigenvalues.

text admin

That geometric rigidity is the reason symmetric matrices are central to the spectral theorem, positive definiteness, Hessian tests, and covariance matrices.

text admin

This page works over the [real numbers](/page/Real%20Numbers). Over complex vector spaces, the closely related analogue is Hermitian symmetry, where transpose is replaced by conjugate transpose; confusing the two loses the inner-product interpretation.

text admin

## Definition

h2 admin

The central condition is that entries mirror across the main diagonal. This is the matrix-level way to say that the interaction from coordinate $i$ to coordinate $j$ is the same as the interaction from coordinate $j$ to coordinate $i$.

text admin

[definition: Symmetric Matrix] Let $n \in \mathbb{N}$. A matrix $A \in \mathbb{R}^{n \times n}$ with entries $A_{ij}$ is symmetric if \begin{align*} A_{ij}=A_{ji} \end{align*} for all $1 \le i,j \le n$. [/definition]

definition admin

This definition is deliberately entrywise: it can be checked without choosing any extra language beyond the matrix entries themselves. Most of the theory, however, is cleaner once the row-column reflection operation has a name.

text admin

## Basic Notation and Related Definitions

h2 admin

The compact test for symmetry is $A^\top=A$, so the transpose operation must be fixed before it is used in later characterisations. Entrywise conditions are good for checking a matrix by hand, but they become awkward once matrices are added, multiplied, or used to represent linear maps. Naming the operation gives a reusable language for every comparison between a matrix and its reflected version.

text admin

[definition: Transpose of a Matrix] Let $m,n \in \mathbb{N}$. The transpose map is the function \begin{align*} T: \mathbb{R}^{m \times n} \to \mathbb{R}^{n \times m}, \qquad A \mapsto A^\top \end{align*} such that, if $A$ has entries $A_{ij}$, then $A^\top$ has entries satisfying \begin{align*} (A^\top)_{ij}=A_{ji}. \end{align*} [/definition]

definition admin

With this notation, a real square matrix is symmetric exactly when $A^\top=A$.

text admin

Once the equation $A^\top=A$ is fixed, it is useful to ask where all matrices satisfying that equation live. They are not scattered examples: adding two such matrices or scaling one preserves the same transpose equation, so the symmetric matrices of a fixed size form a natural ambient space for later projections, bases, and dimension counts.

text admin

[definition: Space of Symmetric Matrices] For $n \in \mathbb{N}$, the space of real symmetric $n \times n$ matrices is \begin{align*} \operatorname{Sym}_n(\mathbb{R}) := \{A \in \mathbb{R}^{n \times n} : A^\top=A\}. \end{align*} [/definition]

definition admin

Symmetry is only one way a matrix can interact simply with transposition. To separate the part unchanged by transpose from the part that cancels under transpose, we also need the opposite sign behavior; these matrices will be invisible to quadratic expressions of the form $x^\top Ax$.

text admin

[definition: Skew-Symmetric Matrix] Let $n \in \mathbb{N}$. A matrix $A \in \mathbb{R}^{n \times n}$ is skew-symmetric if \begin{align*} A^\top=-A. \end{align*} [/definition]

definition admin

In entries, symmetry says $A_{ij}=A_{ji}$ for all $1 \le i,j \le n$.

text admin

Showing 20 of 122 blocks

Verification Progress

122 Total Blocks

0 Verified

0% verified

Contributors

admin 122 blocks (0 verified)

Who Can Verify

Areas: Algebra

Viktor Miykov Admin

Max Vassiliev Global Reviewer

Horia Neagu Global Reviewer

강현욱 Global Reviewer

Demo Testing Global Reviewer

Archie Pennycook Global Reviewer

Quick Actions

Edit Page

Raw Attribution Data

Loading attribution data...

What brings you to Androma?

Start with a route through the knowledge graph.

Attributions & Verification

Content

Verification Progress

Contributors

Who Can Verify

Quick Actions

Sign in to Androma

Check your inbox

One last step

Attributions & Verification

Content

Verification Progress

Contributors

Who Can Verify

Quick Actions

Raw Attribution Data