Matrices

Definition. A matrix of size $m \times n$ is a rectangular table of numbers consisting of $m$ rows and $n$ columns. Matrices are denoted by capital Latin letters, with the dimensions written as a subscript when needed: $A_{m \times n}$ . The entries of a matrix are denoted $a_{ij}$ , where $i$ is the row index (running from $1$ to $m$ ) and $j$ is the column index (running from $1$ to $n$ ).

A = a_{11} a_{21} ⋮ a_{m 1} a_{12} a_{22} ⋮ a_{m 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{mn}

The shorthand notation is $A = (a_{ij})$ , $i = \overline{1, m}$ , $j = \overline{1, n}$ .

An example of a $2 \times 3$ matrix:

A = (1305 - 2 4), a_{11} = 1, a_{13} = - 2, a_{23} = 4.

Definition. Two matrices $A = (a_{ij})$ and $B = (b_{ij})$ are called equal ( $A = B$ ) if they have the same size and all of their corresponding entries coincide: $a_{ij} = b_{ij}$ for all $i, j$ .

Row matrix and column matrix

Definition. A matrix of size $1 \times n$ is called a row matrix (or simply a row), and a matrix of size $m \times 1$ is called a column matrix (or simply a column).

(a_{1} a_{2} \dots a_{n}), b_{1} b_{2} ⋮ b_{m} .

It is precisely columns and rows that play the role of vectors in linear algebra: each is obtained from the other by transposition.

Echelon matrix

Definition. A matrix is called an echelon matrix (one in echelon, or "staircase", form) if two conditions hold:

all zero rows (if any) are placed below all nonzero rows;
the first nonzero entry of every nonzero row, starting from the second, lies strictly to the right of the first nonzero entry of the previous row.

The first nonzero entry of a row is called the leading (or pivot) entry.

An example of an echelon matrix (leading entries in bold):

20001500 0 - 1 00 3470

Echelon form plays a key role in computing the rank of a matrix and in solving systems of linear equations by Gaussian elimination.

Square matrix

Definition. A matrix of size $n \times n$ is called a square matrix of order $n$ . The entries $a_{11}, a_{22}, \dots, a_{nn}$ (those with equal indices) form the main diagonal, and the entries $a_{1 n}, a_{2, n - 1}, \dots, a_{n 1}$ form the anti-diagonal.

A = a_{11} a_{21} ⋮ a_{n 1} a_{12} a_{22} ⋮ a_{n 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{nn}

Triangular matrix

Definition. A square matrix is called triangular if all of its entries on one side of the main diagonal are zero. There are two kinds:

an upper triangular matrix – all entries below the main diagonal are zero ( $a_{ij} = 0$ for $i > j$ );
a lower triangular matrix – all entries above the main diagonal are zero ( $a_{ij} = 0$ for $i < j$ ).

(The entries on the main diagonal itself and on the relevant side of it may be arbitrary – they need not be nonzero.)

200140 - 3 57 (upper), 21 - 3 045007 (lower) .

Property. The determinant of a triangular matrix equals the product of the entries on its main diagonal (see Properties of determinants).

Diagonal matrix

Definition. A diagonal matrix is a square matrix in which every entry off the main diagonal is zero:

D = d_{1} 0 ⋮ 0 0 d_{2} ⋮ 0 \dots \dots ⋱ \dots 00 ⋮ d_{n} = diag (d_{1}, d_{2}, \dots, d_{n}) .

A diagonal matrix is both upper and lower triangular at the same time.

Identity matrix

Definition. The identity matrix $E_{n}$ is the $n \times n$ diagonal matrix all of whose main-diagonal entries equal $1$ :

E_{n} = 10 ⋮ 0 01 ⋮ 0 \dots \dots ⋱ \dots 00 ⋮ 1 = diag (1, 1, \dots, 1) .

Notation: $E$ , $E_{n}$ , $I$ , $I_{n}$ . The identity matrix acts as the neutral element for matrix multiplication: $A \cdot E = A$ and $E \cdot A = A$ (for compatible sizes).

Zero matrix

Definition. The zero matrix $O$ is a matrix all of whose entries are zero.

O = 00 ⋮ 0 00 ⋮ 0 \dots \dots ⋱ \dots 00 ⋮ 0

The zero matrix is the neutral element for addition: $A + O = A$ .

Symmetric and skew-symmetric matrices

Definition. A square matrix $A$ is called symmetric if $a_{ij} = a_{j i}$ for all $i, j$ (that is, $A^{T} = A$ – the matrix is symmetric about its main diagonal).

A = 120 23 - 1 0 - 1 5, a_{ij} = a_{j i} .

Definition. A square matrix $A$ is called skew-symmetric (antisymmetric) if $a_{ij} = - a_{j i}$ for all $i, j$ (that is, $A^{T} = - A$ ). In particular, every entry on its main diagonal is zero.

A = 0 - 2 3 20 - 1 - 3 10, a_{ij} = - a_{j i} .

Symmetric matrices are especially important: quadratic forms lead to them, and their eigenvalues are always real.