Page 1 Matrices and Determinants 1. Matrices 1.1. Basic concepts A matrix is a rectangular array of (real or complex) numbers: A = 2 6 6 6 4 a 11 a 12 a 1m a 21 a 22 a 2m . . . . . . . . . . . . a n1 a n2 a nm 3 7 7 7 5 The numbers in the matrix are called its entries. The size of a matrix is described by the number of its rows and its columns. A has n rows and m columns, thus it is an nm (or: n by m) matrix. Matrices A and B are equal if a ij = b ij for any i and j, and A and B are of the same size. A matrix with just one column is a column vector: b = 2 6 6 6 4 b 1 b 2 . . . b n 3 7 7 7 5 A matrix with just one row is a row vector: c = c 1 c 2 c m A square matrix has an equal number of rows and columns: S = 2 6 6 6 4 a 11 a 12 a 1n a 21 a 22 a 2n . . . . . . . . . . . . a n1 a n2 a nn 3 7 7 7 5 A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes: 0 = 2 6 6 6 4 0 0 0 0 0 0 . . . . . . . . . . . . 0 0 0 3 7 7 7 5 1 Page 2 Matrices and Determinants 1. Matrices 1.1. Basic concepts A matrix is a rectangular array of (real or complex) numbers: A = 2 6 6 6 4 a 11 a 12 a 1m a 21 a 22 a 2m . . . . . . . . . . . . a n1 a n2 a nm 3 7 7 7 5 The numbers in the matrix are called its entries. The size of a matrix is described by the number of its rows and its columns. A has n rows and m columns, thus it is an nm (or: n by m) matrix. Matrices A and B are equal if a ij = b ij for any i and j, and A and B are of the same size. A matrix with just one column is a column vector: b = 2 6 6 6 4 b 1 b 2 . . . b n 3 7 7 7 5 A matrix with just one row is a row vector: c = c 1 c 2 c m A square matrix has an equal number of rows and columns: S = 2 6 6 6 4 a 11 a 12 a 1n a 21 a 22 a 2n . . . . . . . . . . . . a n1 a n2 a nn 3 7 7 7 5 A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes: 0 = 2 6 6 6 4 0 0 0 0 0 0 . . . . . . . . . . . . 0 0 0 3 7 7 7 5 1 A square matrix is called diagonal if all of its entries, apart from the leading diagonal(top left to lower right), are zeroes: D = 2 6 6 6 4 a 11 0 0 0 a 22 0 . . . . . . . . . . . . 0 0 a nn 3 7 7 7 5 A diagonal matrix is called unit matrix if all of its entries in the leading diagonal are 1's (and all of its other entries are zeroes): I = 2 6 6 6 4 1 0 0 0 1 0 . . . . . . . . . . . . 0 0 1 3 7 7 7 5 A square matrix is symmetrical if a ij =a ji for any i and j: F = 2 6 6 6 4 a 11 a 12 a 1n a 12 a 22 a 2n . . . . . . . . . . . . a 1n a 2n a nn 3 7 7 7 5 A square matrix is skewsymmetrical if a ij =a ji for any i and j. In this case any entry in the leading diagonal needs to be 0: G = 2 6 6 6 4 0 a 12 a 1n a 12 0 a 2n . . . . . . . . . . . . a 1n a 2n 0 3 7 7 7 5 2 Page 3 Matrices and Determinants 1. Matrices 1.1. Basic concepts A matrix is a rectangular array of (real or complex) numbers: A = 2 6 6 6 4 a 11 a 12 a 1m a 21 a 22 a 2m . . . . . . . . . . . . a n1 a n2 a nm 3 7 7 7 5 The numbers in the matrix are called its entries. The size of a matrix is described by the number of its rows and its columns. A has n rows and m columns, thus it is an nm (or: n by m) matrix. Matrices A and B are equal if a ij = b ij for any i and j, and A and B are of the same size. A matrix with just one column is a column vector: b = 2 6 6 6 4 b 1 b 2 . . . b n 3 7 7 7 5 A matrix with just one row is a row vector: c = c 1 c 2 c m A square matrix has an equal number of rows and columns: S = 2 6 6 6 4 a 11 a 12 a 1n a 21 a 22 a 2n . . . . . . . . . . . . a n1 a n2 a nn 3 7 7 7 5 A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes: 0 = 2 6 6 6 4 0 0 0 0 0 0 . . . . . . . . . . . . 0 0 0 3 7 7 7 5 1 A square matrix is called diagonal if all of its entries, apart from the leading diagonal(top left to lower right), are zeroes: D = 2 6 6 6 4 a 11 0 0 0 a 22 0 . . . . . . . . . . . . 0 0 a nn 3 7 7 7 5 A diagonal matrix is called unit matrix if all of its entries in the leading diagonal are 1's (and all of its other entries are zeroes): I = 2 6 6 6 4 1 0 0 0 1 0 . . . . . . . . . . . . 0 0 1 3 7 7 7 5 A square matrix is symmetrical if a ij =a ji for any i and j: F = 2 6 6 6 4 a 11 a 12 a 1n a 12 a 22 a 2n . . . . . . . . . . . . a 1n a 2n a nn 3 7 7 7 5 A square matrix is skewsymmetrical if a ij =a ji for any i and j. In this case any entry in the leading diagonal needs to be 0: G = 2 6 6 6 4 0 a 12 a 1n a 12 0 a 2n . . . . . . . . . . . . a 1n a 2n 0 3 7 7 7 5 2 1.2. Operations on Matrices 1.2.1. The Transpose of a Matrix The transpose A T of a matrix A is obtained by interchanging the rows and columns of A. Property: (A T ) T =A. 1.2.2. Addition of Matrices If two matrices are of the same size, they can be added by summing the corres pondig entries. (Subtraction can be dened similarly.) Properties: (a) A +B =B +A (addition is commutative); (b) (A +B) +C =A + (B +C) (addition is associative); (c) (A +B) T =A T +B T . 1.2.3. Multiplication by a Scalar If a ij is a typical entry of A, then A is a matrix, whose corresponding entry is a ij . Properties: (a) ()A =(A); (b) ( +)A =A +A; (c) (A +B) =A +B. 1.2.4. Multiplication of Matrices (a) A row vector and a column vector with the same number of entries can be multiplied as the scalar product is dened: a 1 a 2 a n 2 6 6 6 4 b 1 b 2 . . . b n 3 7 7 7 5 =a 1 b 1 +a 2 b 2 +::: +a n b n (b) The denition can be extended to cover the productAB of any two matri ces, provided that the number of columns in A is the same as the number of rows in B. 3 Page 4 Matrices and Determinants 1. Matrices 1.1. Basic concepts A matrix is a rectangular array of (real or complex) numbers: A = 2 6 6 6 4 a 11 a 12 a 1m a 21 a 22 a 2m . . . . . . . . . . . . a n1 a n2 a nm 3 7 7 7 5 The numbers in the matrix are called its entries. The size of a matrix is described by the number of its rows and its columns. A has n rows and m columns, thus it is an nm (or: n by m) matrix. Matrices A and B are equal if a ij = b ij for any i and j, and A and B are of the same size. A matrix with just one column is a column vector: b = 2 6 6 6 4 b 1 b 2 . . . b n 3 7 7 7 5 A matrix with just one row is a row vector: c = c 1 c 2 c m A square matrix has an equal number of rows and columns: S = 2 6 6 6 4 a 11 a 12 a 1n a 21 a 22 a 2n . . . . . . . . . . . . a n1 a n2 a nn 3 7 7 7 5 A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes: 0 = 2 6 6 6 4 0 0 0 0 0 0 . . . . . . . . . . . . 0 0 0 3 7 7 7 5 1 A square matrix is called diagonal if all of its entries, apart from the leading diagonal(top left to lower right), are zeroes: D = 2 6 6 6 4 a 11 0 0 0 a 22 0 . . . . . . . . . . . . 0 0 a nn 3 7 7 7 5 A diagonal matrix is called unit matrix if all of its entries in the leading diagonal are 1's (and all of its other entries are zeroes): I = 2 6 6 6 4 1 0 0 0 1 0 . . . . . . . . . . . . 0 0 1 3 7 7 7 5 A square matrix is symmetrical if a ij =a ji for any i and j: F = 2 6 6 6 4 a 11 a 12 a 1n a 12 a 22 a 2n . . . . . . . . . . . . a 1n a 2n a nn 3 7 7 7 5 A square matrix is skewsymmetrical if a ij =a ji for any i and j. In this case any entry in the leading diagonal needs to be 0: G = 2 6 6 6 4 0 a 12 a 1n a 12 0 a 2n . . . . . . . . . . . . a 1n a 2n 0 3 7 7 7 5 2 1.2. Operations on Matrices 1.2.1. The Transpose of a Matrix The transpose A T of a matrix A is obtained by interchanging the rows and columns of A. Property: (A T ) T =A. 1.2.2. Addition of Matrices If two matrices are of the same size, they can be added by summing the corres pondig entries. (Subtraction can be dened similarly.) Properties: (a) A +B =B +A (addition is commutative); (b) (A +B) +C =A + (B +C) (addition is associative); (c) (A +B) T =A T +B T . 1.2.3. Multiplication by a Scalar If a ij is a typical entry of A, then A is a matrix, whose corresponding entry is a ij . Properties: (a) ()A =(A); (b) ( +)A =A +A; (c) (A +B) =A +B. 1.2.4. Multiplication of Matrices (a) A row vector and a column vector with the same number of entries can be multiplied as the scalar product is dened: a 1 a 2 a n 2 6 6 6 4 b 1 b 2 . . . b n 3 7 7 7 5 =a 1 b 1 +a 2 b 2 +::: +a n b n (b) The denition can be extended to cover the productAB of any two matri ces, provided that the number of columns in A is the same as the number of rows in B. 3 IfA is of sizenm andB is of sizemp then the product AB is matrix C of sizenp, whosec ij entry is the scalar product of the ith row of A and the jth column of B. AB = 2 6 4 b 11 b 1j b 1p . . . . . . . . . b m1 b mj b mp 3 7 5 2 6 6 6 6 6 6 4 a 11 a 1m . . . . . . a i1 a im . . . . . . a n1 a nm 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 c 11 c 1j c 1p . . . . . . . . . . . . c i1 c ij c ip . . . . . . . . . . . . c n1 c nj c np 3 7 7 7 7 7 7 5 Properties: (a) If AB exists, BA does not necessarily exist. If both AB and BA exist, then in general AB6=BA (multiplication is not commutative); (b) Taking the product ofA and the unit matrix (I) of suitable size, the result is A: A nm I m =A nm =I n A nm ; (c) Multiplcation is associative: (AB)C =A (BC); (d) The distributive laws are valid: (A +B)C =AC +BC and A (B +C) =AB +AC; (e) Transposing reverses the order of products: (AB) T =B T A T : 4 Page 5 Matrices and Determinants 1. Matrices 1.1. Basic concepts A matrix is a rectangular array of (real or complex) numbers: A = 2 6 6 6 4 a 11 a 12 a 1m a 21 a 22 a 2m . . . . . . . . . . . . a n1 a n2 a nm 3 7 7 7 5 The numbers in the matrix are called its entries. The size of a matrix is described by the number of its rows and its columns. A has n rows and m columns, thus it is an nm (or: n by m) matrix. Matrices A and B are equal if a ij = b ij for any i and j, and A and B are of the same size. A matrix with just one column is a column vector: b = 2 6 6 6 4 b 1 b 2 . . . b n 3 7 7 7 5 A matrix with just one row is a row vector: c = c 1 c 2 c m A square matrix has an equal number of rows and columns: S = 2 6 6 6 4 a 11 a 12 a 1n a 21 a 22 a 2n . . . . . . . . . . . . a n1 a n2 a nn 3 7 7 7 5 A zero matrix/zero vector is a matrix/vector, all of whose entries are zeroes: 0 = 2 6 6 6 4 0 0 0 0 0 0 . . . . . . . . . . . . 0 0 0 3 7 7 7 5 1 A square matrix is called diagonal if all of its entries, apart from the leading diagonal(top left to lower right), are zeroes: D = 2 6 6 6 4 a 11 0 0 0 a 22 0 . . . . . . . . . . . . 0 0 a nn 3 7 7 7 5 A diagonal matrix is called unit matrix if all of its entries in the leading diagonal are 1's (and all of its other entries are zeroes): I = 2 6 6 6 4 1 0 0 0 1 0 . . . . . . . . . . . . 0 0 1 3 7 7 7 5 A square matrix is symmetrical if a ij =a ji for any i and j: F = 2 6 6 6 4 a 11 a 12 a 1n a 12 a 22 a 2n . . . . . . . . . . . . a 1n a 2n a nn 3 7 7 7 5 A square matrix is skewsymmetrical if a ij =a ji for any i and j. In this case any entry in the leading diagonal needs to be 0: G = 2 6 6 6 4 0 a 12 a 1n a 12 0 a 2n . . . . . . . . . . . . a 1n a 2n 0 3 7 7 7 5 2 1.2. Operations on Matrices 1.2.1. The Transpose of a Matrix The transpose A T of a matrix A is obtained by interchanging the rows and columns of A. Property: (A T ) T =A. 1.2.2. Addition of Matrices If two matrices are of the same size, they can be added by summing the corres pondig entries. (Subtraction can be dened similarly.) Properties: (a) A +B =B +A (addition is commutative); (b) (A +B) +C =A + (B +C) (addition is associative); (c) (A +B) T =A T +B T . 1.2.3. Multiplication by a Scalar If a ij is a typical entry of A, then A is a matrix, whose corresponding entry is a ij . Properties: (a) ()A =(A); (b) ( +)A =A +A; (c) (A +B) =A +B. 1.2.4. Multiplication of Matrices (a) A row vector and a column vector with the same number of entries can be multiplied as the scalar product is dened: a 1 a 2 a n 2 6 6 6 4 b 1 b 2 . . . b n 3 7 7 7 5 =a 1 b 1 +a 2 b 2 +::: +a n b n (b) The denition can be extended to cover the productAB of any two matri ces, provided that the number of columns in A is the same as the number of rows in B. 3 IfA is of sizenm andB is of sizemp then the product AB is matrix C of sizenp, whosec ij entry is the scalar product of the ith row of A and the jth column of B. AB = 2 6 4 b 11 b 1j b 1p . . . . . . . . . b m1 b mj b mp 3 7 5 2 6 6 6 6 6 6 4 a 11 a 1m . . . . . . a i1 a im . . . . . . a n1 a nm 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 c 11 c 1j c 1p . . . . . . . . . . . . c i1 c ij c ip . . . . . . . . . . . . c n1 c nj c np 3 7 7 7 7 7 7 5 Properties: (a) If AB exists, BA does not necessarily exist. If both AB and BA exist, then in general AB6=BA (multiplication is not commutative); (b) Taking the product ofA and the unit matrix (I) of suitable size, the result is A: A nm I m =A nm =I n A nm ; (c) Multiplcation is associative: (AB)C =A (BC); (d) The distributive laws are valid: (A +B)C =AC +BC and A (B +C) =AB +AC; (e) Transposing reverses the order of products: (AB) T =B T A T : 4 1.3. Applications of Matrices In practice there is often a need for summing some of the entries of a matrix, in terchanging two rows (two columns), etc. Computers use matrixmultiplication for these tasks. 1.3.1. Summing Vectors The entries of the rows of a matrix can be added by postmultiplying it by a columnvector, whose all entries are 1's (by a summing vector). Similarly, by premultiplying the matrix by a rowvector with all of its entries 1's, results in the summing of the entries in the columns. E.g.: A = 1 4 3 2 5 3 ; B = 2 4 1 1 1 3 5 ; C = 1 1 AB = 2 0 ; CA = 3 1 0 : 1.3.2. Permutation Matrices As seen above (in multiplication property b), multiplying a matrix with the unit matrix of suitable size results in no change of the original matrix. If the columns of a unit matrix are interchanged (permutated), and a matrix is postmultiplied by this permutationmatrix, then the suitable columns of the original matrix will be permutated. E.g.: 1 4 3 2 5 3 2 4 0 1 0 0 0 1 1 0 0 3 5 = 3 1 4 3 2 5 : " c 3 " c 1 " c 2 " c 3 " c 1 " c 2 Similarly, if the rows of a unit matrix are interchanged (permutated), and a matrix is premultiplied by this permutationmatrix, then the suitable rows of the original matrix will be permutated. E.g.: r 2 ! r 1 ! 0 1 1 0 1 4 3 2 5 3 = 2 5 3 1 4 3 r 2 r 1 5Read More
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