Since any finite dimensional linear operator can be represented as a matrix, matrix algebra and linear algebra are two sides of the same coin. Properties of linear systems are gleaned from either discipline. The following sections draw on both of these perspectives to examine the basic concepts, numerical techniques, and practical constraints of computational linear algebra.
Assuming the symbols xi represent variables and the symbols aij and bi are complex constants, the following is a system of m linear equations in n unknowns.
| $$\displaystyle{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\cdots+{a}_{1n}{x}_{n} = {b}_{1}$$ $$\displaystyle{a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\cdots+{a}_{2n}{x}_{n} = {b}_{2}$$ | (1) |
| $$\displaystyle\cdots$$ | |
| $$\displaystyle{a}_{m1}{x}_{1}+{a}_{m2}{x}_{2}+\cdots+{a}_{mn}{x}_{n} = {b}_{m}$$ |
This system of equations is expressed in matrix notation as
| $$\mathbf{Ax=b}$$ | (2) |
where
| $$\mathbf{A}=\left(\begin{array}[]{cccc}{a}_{11}&{a}_{12}&\cdots&{a}_{1n}\\ {a}_{21}&{a}_{22}&\cdots&{a}_{2n}\\ \cdots&\cdots&\cdots&\cdots\\ {a}_{m1}&{a}_{m2}&\cdots&{a}_{mn}\end{array}\right)\ \mathbf{x}=\left(\begin{array}[]{c}{x}_{1}\\ {x}_{2}\\ \cdots\\ {x}_{n}\end{array}\right)\ \mathbf{b}=\left(\begin{array}[]{c}{b}_{1}\\ {b}_{2}\\ \cdots\\ {b}_{n}\end{array}\right)$$ | (3) |
A rectangular array of coefficients such as A is referred to as a matrix. The matrix A has m rows and n columns. As such, it is called an m × n matrix. A square matrix has an equal number of rows and columns, e.g. an n × n matrix. A vector is a matrix with just one row or just one column. A 1 × n matrix is a row vector. An m × 1 matrix, such as x or b in Equation 2, is called a column vector.
The elements of a matrix aii whose row and column index are equal are referred to as its diagonal. The elements of a matrix above the diagonal (aij, where i < j) are its superdiagonal entries. The elements of a matrix below the diagonal (aij, where i > j) are its subdiagonal entries. A matrix whose subdiagonal entries are zero is called upper triangular. An upper triangular matrix with ones along the diagonal is called unit upper triangular. The following 3 × 3 matrix is unit upper triangular.
| $$\left(\begin{array}[]{ccc}1&{a}_{12}&{a}_{13}\\ 0&1&{a}_{23}\\ 0&0&1\end{array}\right)$$ |
Similarly, a matrix whose superdiagonal entries are zero is called lower triangular. A lower triangular matrix with ones along the diagonal is called unit lower triangular. The following 3 × 3 matrix is lower triangular.
| $$\left(\begin{array}[]{ccc}{a}_{11}&0&0\\ {a}_{21}&{a}_{22}&0\\ {a}_{31}&{a}_{32}&{a}_{33}\end{array}\right)$$ |
A matrix whose superdiagonal and subdiagonal entries are zero is a diagonal matrix, e.g.
| $$\left(\begin{array}[]{ccc}{a}_{11}&0&0\\ 0&{a}_{22}&0\\ 0&0&{a}_{33}\end{array}\right)$$ |
A square matrix whose subdiagonal elements are the mirror image of its superdiagonal elements is referred to as a symmetric matrix. More formally, a symmetric matrix A has the property aij = aji. A trivial example of a symmetric matrix is a diagonal matrix. The general case of a 3 × 3 symmetric matrix follows.
| $$\left(\begin{array}[]{ccc}{a}_{11}&{a}_{12}&{a}_{13}\\ {a}_{12}&{a}_{22}&{a}_{23}\\ {a}_{13}&{a}_{23}&{a}_{33}\end{array}\right)$$ |
A matrix whose elements are all zero is called the zero matrix or the null matrix.