matrix (mathematics)

Matrix

\begin{align*} \begin{pmatrix} a_{11} & b_{12} & \cdots & a_{1n}\\ a_{21} & b_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{pmatrix} \end{align*}

Matrices are incredibly useful, if one can decode what they actually mean. Otherwise, they're just arrays of numbers with some funny rules attached. It can be used to store data, mathematical structures, etc. However, the most common usage is to represent linear transformations and systems of equations, which is also why people often conflate the two concepts.

E.g. a system of equation, say \[2x+y=3,4x-y=1\] can be represented as \[A\mathbf{x}=b\], where \[A=\begin{pmatrix} 2&1\\4&-1 \end{pmatrix}\], \[\mathbf{x}=\begin{pmatrix} x\\y \end{pmatrix}\] and \[b=\begin{pmatrix} 3\\1 \end{pmatrix}\]. In this context, one can think of matrices as a fancy and compact way to write down systems of equations. Though, this fancy representation has many properties encoded in itself to make solving systems of linear equations much more efficient, hence the usefulness of matrices.

Additionally, when we encounter a matrix, e.g. \[\begin{pmatrix} 1&2\\3&8 \end{pmatrix}\], the standard view would be that it represents the points \[(1,3)\] and \[(2,8)\], while if we were to turn that into a representation of a system of linear equations, that would be \[\begin{pmatrix} 1&2\\3&8 \end{pmatrix}\begin{pmatrix} x\\y \end{pmatrix}\], which represents \[x+2y\] and \[3x+8y\].

An \[n\times m\] matrix \[A\] is a rectangular array of numbers with \[n\] rows and \[m\] columns. By \[A=(a_{ij})\] we mean that \[a_{ij}\] is the entry in the \[i\]th row and \[j\]th column. For example, \[A=\begin{pmatrix} 1&2&-1\\0&-1&4 \end{pmatrix}\] is a \[2\times3\] matrix while \[a_{12}=2\]. For matrices with real entries we denote them by \[\mathbb{R}^{n\times m}\].

It is generally acceptable to use both parentheses and square brackets for matrices.

Notations

An \[n\times 1 \] matrix is called a column vector while a \[1\times m\] matrix is called a row vector. An \[n\times n\] matrix is called square. An \[n\times n\] matrix \[A=(a_{ij})\] is known as a diagonal matrix if entries outside the main diagonal are all zero, i.e. \[a_{ij}=0\] for \[i\ne j\], e.g. \[A=\begin{pmatrix} -7&0&0\\0&3&0\\0&0&5\end{pmatrix}\]. Additionally, since \[A\] has dimensions of \[n\times n\], it can also be classified as a square matrix.

The transpose of the \[n\times m\] matrix \[A=(a_{ij})\] is the \[m\times n\] matrix \[A^{T}=(a_{ji})\], e.g. \[A=\begin{pmatrix} 1&2&-2\\0&-1&4 \end{pmatrix}\], then \[A^{T}=\begin{pmatrix} 1&0\\2&-1\\-2&4 \end{pmatrix}\]. It follows that \[\left( A^{T} \right)^{T}=A\].

We denote sets of real coordinates by column vectors. For example, instead of the \[(2,1)\] we're used to, we write \[x=\begin{pmatrix} 2\\1 \end{pmatrix}\]. To save space, some authors might write it as \[(2,1)^{T}\].

If \[A=(a_{ij})\] and \[B=(b_{ij})\] are both \[n\times m\] matrices, then \[A+B\] is the \[n\times m\] matrix with \[(A+B)_{ij}=a_{ij}+b_{ij}\]. Similarly, by \[A-B\] we mean \[A+(-1)B\].

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