Matrices and matrix operations

1. Matrix

Let $m,n$ be positive integers. An $m\times n$ matrix with real entries is an array of $mn$ real numbers arranged in $m$ rows and $n$ columns:

The number $a_{ij}$ is the entry in position $(i,j)$; the first index indicates the row and the second the column. One writes briefly $A=(a_{ij})$, and the set of $m\times n$ matrices is denoted by $M_{m,n}(\mathbb{R})$. Two matrices are equal if they have the same dimensions and the entries in the same position coincide, so an equality between $m\times n$ matrices is equivalent to $mn$ equalities between numbers. Matrices collect into a single object the coefficients of a linear system, of a linear map or of a quadratic form, and the operations between matrices translate the operations on these objects.

2. Square matrices, row vectors and column vectors

• A matrix with $m=n$ is called square of order $n$; the set of square matrices of order $n$ is denoted by $M_n(\mathbb{R})$.
• In a square matrix the entries $a_{11},a_{22},\dots ,a_{nn}$ form the main diagonal. Their sum is the trace, $\mathrm{tr}A=a_{11}+a_{22}+\cdots +a_{nn}$.
• A $1\times n$ matrix is a row vector; an $m\times 1$ matrix is a column vector. In matrix computations the vectors of ${\mathbb{R}}^n$ are written as columns.
• The zero matrix $O$ has all entries equal to $0$.

For $A=\left(\begin{array}{cc}2& 5\\ -1& 3\end{array}\right)$ the main diagonal consists of $2$ and $3$, therefore $\mathrm{tr}A=5$.

1. Diagonal, triangular and identity matrices

Let $A=(a_{ij})\in M_n(\mathbb{R})$.

• $A$ is upper triangular if $a_{ij}=0$ for $i>j$, that is, if the entries below the main diagonal are zero.
• $A$ is lower triangular if $a_{ij}=0$ for $i<j$, that is, if the entries above the main diagonal are zero.
• $A$ is diagonal if $a_{ij}=0$ for $i\ne j$; one writes $A=\mathrm{diag}(a_{11},\dots ,a_{nn})$.
• The identity matrix $I_n$ is the diagonal matrix with all diagonal entries equal to $1$.

A diagonal matrix is both upper and lower triangular. The product of two upper triangular matrices of the same order is upper triangular, and the same holds for diagonal matrices. On these classes computations become simpler: row echelon reduction produces upper triangular matrices, and the powers of a diagonal matrix are computed entry by entry.

2. Symmetric and skew-symmetric matrices

A matrix $A=(a_{ij})\in M_n(\mathbb{R})$ is called

• symmetric if $a_{ij}=a_{ji}$ for all $i,j$;
• skew-symmetric if $a_{ij}=-a_{ji}$ for all $i,j$.

In a symmetric matrix the entries are arranged as mirror images with respect to the main diagonal. In a skew-symmetric matrix, setting $i=j$ gives $a_{ii}=-a_{ii}$, therefore the diagonal entries are zero. Symmetric matrices describe quadratic forms and conics, and the spectral theorem guarantees that they are always diagonalizable; these are covered in the topics on the inner product and on conics.