Matrix (mathematics)

A matrix - is rectangular array of elements. An entry in the form of a rectangular table containing m-rows and n-columns is called a matrix and is written in the form:

$$ \left| \begin{array}{rrr} 0 & -33 & 58 & 45 \\ 23 & 0 & 7 & 5 \\ 6 & 0 & -11 & 21 \end{array}\right|, \ \left[ \begin{array}{rrr} a & f \\ c & d \end{array}\right], \ \left( \begin{array}{ccc} \cos\alpha \\ sin\alpha \\ sin(\alpha + \beta) \end{array}\right) $$

for the first matrix: $$ \left( \begin{array}{rrr} 0 & -33 & 58 & 45 \\ 23 & 0 & 7 & 5 \\ 6 & 0 & -11 & 21 \end{array}\right) $$

• is a 3×4 matrix
• three rows: $ \left( \begin{array}{ccc} 0 & -33 & 58 & 45 \end{array}\right), \ \left( \begin{array}{ccc} 23 & 0 & 7 & 5 \end{array}\right), \ \left( \begin{array}{ccc} 6 & 0 & -11 & 21 \end{array}\right), \ $
• four columns: $ \left( \begin{array}{ccc} 0 \\ 23 \\ 6 \end{array}\right), \ \left( \begin{array}{ccc} -33 \\ 0 \\ 0 \end{array}\right), \ \left( \begin{array}{ccc} 58 \\ 7 \\ -11 \end{array}\right), \ \left( \begin{array}{ccc} 45 \\ 5 \\ 21 \end{array}\right) $
• its elements are: $ a_{11}\!=\!0,\ a_{12}\!=\!-33,\ a_{13}\!=\!58,\ a_{14}\!=\!45,$ $ a_{21}\!=\!23,\ a_{22}\!=\!0,\ a_{23}\!=\!7,\ a_{24}\!=\!5,\ $ $ a_{31}\!=\!6,\ a_{32}\!=\!0,\ a_{33}\!=\!-11,\ a_{34}\!=\!21 $

It is convenient to think of every element of a matrix as belonging to a certain row and a certain column of the matrix. If a matrix has m - rows and n - columns, the matrix is said of order m by n, or m × n. Every element of a matrix can be uniquely characterized by a row index and a column index.

It is convenient to write a m × n matrix as:

$$ A = \left[ \begin{array}{rrr} a_{11} & a_{12} & a_{13} & ... & a_{1n} \\ a_{21} & a_{22} & a_{23} & ... & a_{2n} \\ a_{31} & a_{32} & a_{33} & ... & a_{3n} \\ ... & ... & ... & ... & ... \\ a_{m1} & a_{m2} & a_{m3} & ... & a_{mn} \end{array}\right] $$

where $a_{ij}$ is the element of i-th row and j-th column, it may be real or complex number or functions. The elements may even be matrices themselves, in which case the elements are called submatrices and the whole matrix is said to be partitioned.

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Writing data in the form of a matrix allows you to compactly provide a set of data (numbers, symbols, variables, etc.) and subsequently perform mathematical operations on these data or mathematical transformations of the data recorded in the matrix.

Using matrices, problems are solved in three categories:

• in the theory of transformation
• in the solution of systems of linear equations
• in the solution of eigenvalue problems

Some Special Matrices

Zero Matrix

A matrix of arbitrary order is said to be a zero matrix if and only if every element of the matrix equals zero. A zero matrix is sometimes called a null matrix.

$ \left( \begin{array}{rrr} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) $

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Row Matrix

A row matrix has only one row. A row matrix is also called a row vector. If it is called row vector, the elements of the matrix are usually referred as components.

$ \left( \begin{array}{rrr} 3x & 2y & 45xy & 9 \end{array}\right) $

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Column Matrix

A column matrix has only one column. A column matrix is also called a column vector. Again if it is called column vector, the elements of the matrix are usually called the components of the vector.

$ \left( \begin{array}{ccc} 45 \\ 3 \\ 0 \end{array}\right) $

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Square Matrix

A matrix is said to be a square matrix if the number of rows equals the number of columns. A square matrix of order n simply means it has n rows and n columns. Square matrix is of particular importance. We will be dealing mostly with square matrices together with column and row matrices.

For a square matrix A, we can calculate the determinant

$$ det (A) = |A| $$

as defined in Chap. 4. Matrix is not a determinant. Matrix is an array of numbers, determinant is a single number. The determinant of a matrix can only be defined for a square matrix.

Let $ A = (a_{ij})_{n} $ be a square matrix of order $n$. The diagonal going from the top left corner to the bottom right corner of the matrix, its elements $ a_{11}, a_{22}, ... , a_{nn} $, are called the diagonal elements. All the remaining elements aij for $ i \ne j $ are called the off-diagonal elements.

There are several special square matrices that are of interest.

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Diagonal Matrix

A diagonal matrix is a square matrix whose diagonal elements are not all equal to zero, but off-diagonal elements are all zero. For example,

$ \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -2 \end{array}\right) \quad and \quad \left( \begin{array}{rrr} 3 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 5 \end{array}\right) $

are diagonal matrices. Therefore for a diagonal matrix $ (A)_{ij} = a_{ii} \cdot δ_{ij}, $ where $ \text δ_{ij} = \Bigl\{ \begin{array}{ccc} 1 \; i\!=\!j, \\ 0 \; i\!\ne\!j. \end{array} $

This kind of notation may seem to be redundant, as a diagonal matrix can easily be visualized. However, this notation is useful in manipulating matrices.

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Constant Matrix

If all elements of a diagonal matrix happen to be equal to each other, it is said to be a constant matrix or a scalar matrix.

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Unit Matrix

If the elements of a constant matrix are equal to unity, then it is a unit matrix. A unit matrix is also called the Identity matrix, denoted by I, that is

$ I = \left( \begin{array}{ccc} 1 & 0 & ... & 0 \\ 0 & 1 & ... & 0 \\ ... & ... & ... & ... \\ 0 & 0 & ... & 1 \end{array}\right) $

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Triangular Matrix

A square matrix having only zero elements on one side of the principal diagonal is a triangular matrix. Thus

$ A = \left( \begin{array}{rrr} 1 & 2 & 3 \\ 0 & 3 & 4 \\ 0 & 0 & -2 \end{array}\right), \quad B = \left( \begin{array}{rrr} 1& 0 & 0 \\ 3 & 2 & 0 \\ 4 & 5 & 0 \end{array}\right), \quad C = \left( \begin{array}{rrr} 0& 0 & 0 \\ 5 & 0 & 0 \\ 4 & 3 & 0 \end{array}\right) $

are examples of a triangular matrix. A matrix for which $ a_{ij} = 0$ for $i > j $ is called a right-triangular matrix or a upper triangular matrix, such as matrix A above. Whereas a matrix with $a_{ij} = 0$ for $i < j$ is called a left-triangular matrix or a lower triangular matrix, such as matrix B. If all the principal diagonal elements are zero, the matrix is a strictly triangular matrix, such as matrix C. Diagonal matrix, identity matrix as well as zero matrix are all triangular matrices.