Introductory Linear Algebra/Matrices
Motivation[edit | edit source]
One important application for matrices is solving systems of linear equations. Some of the following definitions may be viewed as 'designed for solving system of linear equations'.
Some terminologies[edit | edit source]
(Matrix) A matrix (plural: matrices) is a rectangular array of numbers. A horizontal unit is a row, and a vertical unit is a column. The element in the th row and the th column is the th entry of the matrix.
An (read 'm by n') is a matrix with rows and columns, and is the size of the matrix. The rows are counted from the top, and the columns are counted from the left. If the size of a matrix is , we simply refer to this matrix as a number, and no brackets are needed in this case. The set of all matrices with real entries is denoted by . A capital letter is usually used to denote a matrix, while small letters are used to denote its entries. For example, denotes an matrix with entries in which and . (We may omit the subscript specifying the size of matrix if its size is already mentioned, or its size is not important.)
Definition. (Matrix equality) Two matrices and are equal if
- for each pair .
We write if and are equal.
- In other words, if two matrices have the same size and same entries, then they are equal.
- If and are not equal, we write .
Exercise. Consider the following three matrices and .
Exercise. Let be a matrix in which each entry . Write down in the form of an array of numbers.
In particular, if a matrix has the same number of rows and columns, then it has some nice properties. In view of the shape of such a matrix (square-like), we define such matrices as square matrices.
Definition. (Square matrix) A square matrix is a matrix with the same number of rows and columns.
We will also introduce a term, namely main diagonal, which will be useful in some situations.
Definition. (Main diagonal) The main diagonal of an matrix (which is a square matrix) is the collection of the th, th, , th entries.
Example. The main diagonal of the matrix is the collection of and .
Remark. The matrix is the identity matrix (which will be defined later).
Then, we will define some types of matrices for which the definitions are related to the main diagonal.
Definition. (Triangular matrix) A triangular matrix is an upper triangular matrix or a lower triangular matrix (inclusively).
An upper triangular matrix is a square matrix whose entries below its main diagonal are all .
A lower triangular matrix is a square matrix whose entries above its main diagonal are all .
- Equivalently and symbolically, a matrix is upper triangular if whenever ,
and is lower triangular if if .
- Upper triangular and lower triangular matrices are in the form of
respectively, in which is an arbitrary entry (which may or may not be zero).
Definition. (Diagonal matrix) A diagonal matrix is a square matrix whose entries not lying on the main diagonal are all .
- A diagonal matrix is both upper triangular and lower triangular.
- A diagonal matrix has the form
in which is an arbitrary entry.
The last terminology we mention here is submatrix, which will sometimes be used.
Definition. (Submatrix) Let be a matrix. A submatrix of is a matrix obtained from by removing some rows or columns (inclusively).
Remark. By convention, every matrix is a submatrix of itself.
Matrix operations[edit | edit source]
In this section, we will cover different matrix operations. Some operations are quite different from that in the number system, in particular, matrix multiplication.
Definition. (Matrix addition and subtraction) Let and be two matrices of the same size. We define matrix addition and subtraction by
Definition. (Scalar multiplication of matrix) Let be a matrix. We define the scalar multiplication of the matrix by
Then, we are going to define matrix multiplication, which is quite different from the multiplication in the number system.
(Matrix multiplication) Let and be two matrices. The matrix product of and is defined as the matrix whose th entry is
On the other hand, a positive power of a square matrix is defined quite similarly to that in number system.
Definition. (Positive power of a square matrix) Let be a square matrix. The th power of , written , in which is a positive number, is the product of copies of , i.e.,
Then, we will discuss matrix analogs for the numbers zero and one in the number system, namely the zero matrix and the identity matrix, which, in the number system, are analogous to the numbers and respectively.
Definition. (Zero matrix) The zero matrix is the matrix whose entries are all , and is denoted by or simply if there is no ambiguity.
Remark. The zero matrix is analogous to the number in the number system, because:
- We have for each matrix of the same size as the zero matrix.
- We have if the products are well-defined, for each matrix .
Definition. (Identity matrix) The identity matrix, denoted by or simply if there is no ambiguity, is the diagonal matrix whose diagonal entries are all .
Remark. The identity matrix is analogous to the number in the number system, because if the products are well-defined, for each matrix .
- the zero matrix is
- the identity matrix is
Proposition. (Properties of matrix operations) Let and be matrices such that the following operations are well-defined, and let be a scalar. Then, the following hold.
(i) (associativity of matrix multiplication) .
(ii) (roles of numbers zero and one)
(iii) (distributivity of matrix multiplication)
Remark. Matrix multiplication is not commutative in general, i.e., the matrix product is different from the matrix product in general.
Then, we will introduce an operation that does not exist in the number system, namely transpose.
Definition. (Matrix transpose) Let be a matrix. The transpose of matrix is the matrix
Remark. We can see from the definition that the transpose of matrix is simply itself.
Let be the matrix . Then,
Proposition. (Properties of matrix transpose) Let and be matrices such that the following operations are well-defined. Then, the following hold.
(ii) (linearity) for each real number and
(iii) ('reverse multiplicativity')
(Symmetric matrix) A matrix is a symmetric matrix if
Definition. (Skew-symmetric matrix) A matrix is a skew-symmetric matrix if
Proposition. (Necessary condition for a symmetric and a skew-symmetric matrix) A symmetric matrix or a skew-symmetric matrix must be a square matrix.
Proof. It follows from observing that matrix transpose has the same size as the original matrix if and only if the matrix is square matrix, since the number of rows and number of columns are swapped for the matrix transpose, and the size remains unchanged if and only if the number of rows equals number of columns.
Remark. This does not imply that every square matrix is a symmetric matrix or a skew-symmetric matrix. The condition that a matrix is a square matrix itself is not sufficient (but necessary) for it to be symmetric or skew-symmetric.
Example. is symmetric, while is skew-symmetric, and its transpose is . In a skew-symmetric matrix, all entries lying on the main diagonal must be 0.