A symmetric matrix is a specific subset of square matrices that are perfectly equal to their own transpose. Mathematically, a square matrix $$\mathbf{A}$$ is considered symmetric if and only if the condition $$\mathbf{A} = \mathbf{A}^	op$$ holds true. This implies that the matrix elements are mirror images across the main diagonal, meaning that $$a_{ij} = a_{ji}$$ for all valid row indices $$i$$ and column indices $$j$$.

Symmetric Matrix

A matrix is classified as a square matrix when its number of horizontal rows, $$m$$, is exactly equal to its number of vertical columns, $$n$$ (i.e., $$m = n$$).

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The order of a matrix is defined by its dimensions, specifically its number of rows and columns. A matrix possessing $$m$$ rows and $$n$$ columns has an order of $$m 	imes n$$, which is often read as an '$$m$$ by $$n$$' matrix. For instance, a matrix containing $$2$$ rows and $$3$$ columns is designated as a $$2 	imes 3$$ matrix.

Order of a Matrix

In mathematical notation, matrices are conventionally denoted by bold capital letters, such as $$\mathbf{X}$$, $$\mathbf{Y}$$, and $$\mathbf{Z}$$. To specify the dimensions and the set from which the elements are drawn, the notation $$\mathbf{A} \in \mathbb{R}^{m 	imes n}$$ is used. This expression formally indicates that the matrix $$\mathbf{A}$$ is composed of $$m 	imes n$$ real-valued scalars, logically arranged into exactly $$m$$ rows and $$n$$ columns.

Typographical Conventions for Matrices

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Dive into Deep Learning

Square Matrix

In deep learning frameworks, a mathematical matrix $$\mathbf{A} \in \mathbb{R}^{m \times n}$$ is programmatically implemented as a $$2^{\textrm{nd}}$$-order tensor possessing a shape of $$(m, n)$$. You can dynamically construct a matrix from an existing appropriately sized tensor by utilizing the `reshape` function (e.g., `.reshape(m, n)`). This operation reorganizes the sequential data into the specified row and column dimensions, allowing a generic one-dimensional tensor of $$m \times n$$ elements to be explicitly cast into a valid $$m \times n$$ matrix structure without altering the underlying numerical values.

Programmatic Representation of Matrices

To refer to a specific element within a matrix, mathematical notation utilizes a lowercase letter corresponding to the matrix name, accompanied by subscripts denoting its exact position. For instance, $$a_{ij}$$ represents the specific scalar value located in the $$i^{	extrm{th}}$$ row and the $$j^{	extrm{th}}$$ column of the matrix $$\mathbf{A}$$. This notation allows for the precise algebraic definition of the entire matrix structure, which is visually illustrated as a tabular grid of these subscripted elements:

$$ \mathbf{A}=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

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