MATHEMATICS 1 Applied Informatics 3 Štefan BEREŽNÝ Contents LINEAR ALGEBRA Matrices Determinants
MATHEMATICS 1 Applied Informatics 4 Štefan BEREŽNÝ Matrix Definition: A rectangular array of m n real numbers written in m rows and n columns is called a matrix of the type m n (read: type m by n or shortly an m by n matrix). The numbers which are contained in the matrix are called its entries or its elements. Matrices are usually denoted by capital letters and their entries are denoted by the same small letters with two indices. The indices are related to the position of the entry a ij ( i -th row and j -th column in matrix A ).
MATHEMATICS 1 Applied Informatics 5 Štefan BEREŽNÝ Matrix Specially type of matrices: (a)Upper triangular matrix: If all elements under the main diagonal are equal to zero, then matrix A is called the upper triangular matrix. (b)Lower triangular matrix: If all elements above the main diagonal are equal to zero, then matrix A is called the lower triangular matrix. (c)Zero matrix: A matrix whose all elements are equal to zero is called zero matrix.
MATHEMATICS 1 Applied Informatics 6 Štefan BEREŽNÝ Matrix Specially type of matrices: (d)Transposed matrix: The n m matrix B = b ij whose elements satisfy b ij = a ji for i = 1, 2,..., m and j = 1, 2,..., n is called a transposed matrix to matrix A. It is denoted by A T. In other words: the transposed matrix A T to matrix A can be obtained by turning A over the main diagonal.
MATHEMATICS 1 Applied Informatics 7 Štefan BEREŽNÝ Matrix Specially type of matrices: (e)Square matrix: A matrix with the same number of rows as columns is said to be a square matrix. (f)Diagonal matrix: The square n n matrix that is upper and lower triangular matrix together is called diagonal matrix. It is denoted by diag( A ) = a 11, a 22, a 33, …, a nn .
MATHEMATICS 1 Applied Informatics 8 Štefan BEREŽNÝ Matrix Specially type of matrices: (g)Identity matrix: The diagonal matrix A whose elements equal 1 is called the identity matrix. It is denoted by E or I. (h)Symmetric matrix: The square matrix A is symmetric if satisfy: A = A T.
MATHEMATICS 1 Applied Informatics 9 Štefan BEREŽNÝ Matrix Two matrices are identical if they are of the same type and if they have the same entries at corresponding position. Suppose that A = a ij is an m n matrix. The entries a 11, a 22, a 33, …, a kk (where k = min m, n ) form a so called main diagonal in matrix A.
MATHEMATICS 1 Applied Informatics 10 Štefan BEREŽNÝ Matrix operations Addition of matrices: If matrices A = a ij and B = b ij are both m n then their sum is the m n matrix C = c ij with elements c ij = a ij + b ij for i = 1, 2, …, m and j = 1, 2, …, n. We use the notation C = A + B.
MATHEMATICS 1 Applied Informatics 11 Štefan BEREŽNÝ Matrix operations Multiplication of matrices by real numbers: If A = a ij is an m n matrix and R, then the product of the number and matrix A is the matrix C = c ij of the same type m n with elements c ij = a ij for i = 1, 2, …, m and j = 1, 2, …, n. We say: matrix C is -multiple of matrix A. We use the notation C = A or C = A. Matrices of the same type can also be subtracted. The difference of matrices A and B is the matrix C = A + ( 1) B = A B.
MATHEMATICS 1 Applied Informatics 12 Štefan BEREŽNÝ Matrix operations Multiplication of matrices: If A = a ij is an m k matrix and B = b ij is k n then the product of the matrices A and B is the m n matrix C = c ij whose elements satisfy:
MATHEMATICS 1 Applied Informatics 13 Štefan BEREŽNÝ Matrix operations for i = 1, 2, …, m and j = 1, 2, …, n. We write: C = A B. You can observe that the element c ij in matrix C is the scalar product of i -th row of matrix A with j -th column of matrix B. Holds:Multiplication of matrices is not commutative!
MATHEMATICS 1 Applied Informatics 14 Štefan BEREŽNÝ Matrix operations Rules for operations with matrices: (1) A + B = B + A, (2)( A + B ) + C = A + ( B + C ), (3)( A B ) C = A ( B C ), (4) ( A + B ) = A + B, (5)( + ) A = A + A,
MATHEMATICS 1 Applied Informatics 15 Štefan BEREŽNÝ Matrix operations Definition: The maximum number of linearly independent rows (or linearly independent columns) is called the rank of matrix A. We denote it rank( A ) or r( A ). Rows and columns are taken as arithmetic vectors. Theorem: Let A be an m n upper triangular matrix and let all elements on the main diagonal be different from zero. Then the rank of matrix A is equal to the minimum of the numbers m and n.
MATHEMATICS 1 Applied Informatics 16 Štefan BEREŽNÝ Matrix operations Elementary row and column operations: We can transform the non-upper triangular matrix to an upper triangular matrix using so called elementary row and column operations, which do not change the rank of the matrix. We shall use the following elementary row operations:
MATHEMATICS 1 Applied Informatics 17 Štefan BEREŽNÝ Matrix operations (a)change of order of rows, (b)multiplication of some row by a nonzero real number, (c)addition to some row of a linear combination of the other rows, (d)omission of a row which is a linear combination of the other rows.
MATHEMATICS 1 Applied Informatics 18 Štefan BEREŽNÝ Matrix operations All the operations can also be performed with columns. The procedure of transformation of an arbitrary matrix to an upper triangular matrix (all of whose elements on the main diagonal are different from zero) by means of the elementary row and column operations is called the Gauss algorithm.
MATHEMATICS 1 Applied Informatics 19 Štefan BEREŽNÝ Determinant Definition: Let A be a square matrix. The determinant of matrix A is the number which is denoted by det( A ) and which is assigned to matrix A in accordance with these rules: (a)If A = a is a 1 1 square matrix then det( A ) = a. (b)If A = a ij is an n n square matrix (for n 1) then we choose an arbitrary i -th row of matrix A and we put: det( A ) = a i1 A i1 + a i2 A i2 + a i3 A i3 + … + a in 1 A in 1 + a in A in where A ij is a so called co-factor of element a ij.
MATHEMATICS 1 Applied Informatics 20 Štefan BEREŽNÝ Determinant The co-factor is equal to ( 1) i+j det( A ij ) where det( A ij ) is the determinant of the ( n 1) ( n 1) square matrix which arises from A by omission the i -th row and the j - th column. det( A ij )is called the minor, which is the abbreviation for “minor determinant”. The sum a i1 A i1 + a i2 A i2 + a i3 A i3 + … + a in 1 A in 1 + a in A in is called the expansion of the determinant according to the i -th row. The expansion of the determinant according to the j -th column is: det( A ) = a 1j A 1j + a 2j A 2j + a 3j A 3j + … + a n 1j A n 1j + a nj A nj.
MATHEMATICS 1 Applied Informatics 21 Štefan BEREŽNÝ Determinant Saruss' rule: The determinant of a 2 2 and 3 3 matrix can also be, apart from the expansion according to some row or column, computed by the so called “Saruss' rule”: det( A ) = a 11 a 22 + a 12 a 21 and det( A ) = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 ( a 13 a 22 a 31 + a 11 a 23 a 32 + a 12 a 21 a 33 ).
MATHEMATICS 1 Applied Informatics 22 Štefan BEREŽNÝ Determinant Important facts about determinants: (1)If all elements in some row or column of matrix A are zero then det( A ) = 0. (2)Interchanging two rows or columns changes the sign of the determinant. (3)If two rows or columns are identical the determinant is zero. (4)If we multiply some row or column of matrix A by a real number then the determinant of the new matrix is equal to det( A ).
MATHEMATICS 1 Applied Informatics 23 Štefan BEREŽNÝ Determinant Important facts about determinants: (5)If any row respectively column of matrix A is a multiple of another row respectively column of A, the determinant of A is zero. (6)If any row respectively column of matrix A is a linear combination of the other rows respectively columns of matrix A, the determinant is zero. (7)det( A ) = det( A T ). (8)If A and B are n n square matrices then det( A B ) = det( A ) det( B ).
MATHEMATICS 1 Applied Informatics 24 Štefan BEREŽNÝ Determinant Definition: An n n square matrix which has the maximum possible rank is called a regular matrix. (i.e. rank( A ) = n ) Definition: Suppose that A is an n n square matrix and E is the n n identity matrix. An n n square matrix A 1 is called the inverse matrix to matrix A if: A A 1 = A 1 A = E.
MATHEMATICS 1 Applied Informatics 25 Štefan BEREŽNÝ Determinant Theorem: Let A be a square matrix. Then the following statements are equivalent: (1) A is regular. (2)det( A ) 0. (3)The inverse matrix A 1 exists.
MATHEMATICS 1 Applied Informatics 26 Štefan BEREŽNÝ Determinant Theorem: If A and B are n n regular matrices then matrix A B is also regular matrix. Moreover, it holds: ( A B ) 1 = B 1 A 1. Theorem: If matrix A is regular then matrix A 1 is also regular. Moreover, it holds: (1)( A 1 ) 1 = A, (2) A A 1 = A 1 A = E.
MATHEMATICS 1 Applied Informatics 27 Štefan BEREŽNÝ Determinant Theorem: (Uniqueness of the inverse matrix) If a square matrix A has an inverse matrix then the inverse matrix A 1 is unique. Let A is an n n square regular matrix. For inverse matrix A 1 to matrix A holds:
MATHEMATICS 1 Applied Informatics 28 Štefan BEREŽNÝ Determinant The adjoint matrix Adj( A ) of matrix A is the transpose matrix of the matrix of co-factors of the elements a ij in matrix A.
MATHEMATICS 1 Applied Informatics 29 Štefan BEREŽNÝ Thank you for your attention.