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Matrix Algebra FormulasMatrix Algebra Formulas "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #2 20060411 15:11:14
Re: Matrix Algebra FormulasMatrices, Introduction and different types An array of mn numbers written in m rows and n columns is called a matrix of order m x n. If in a matrix, m ≠ n, then it is a rectangular matrix. If in a matrix, m=n, then it is a square matrix. If a matrix contains only one row, (i.e. 1xn matrix), then it is a row matrix. If a matrix contains only 1 column, (i.e. mx1 matrix), then it is a column matrix. A matrix whose every element is zero is a zero matrix. A square matrix is called a diagonal matrix if all its elements other than the elements in the leading diagonal are zero. A square matrix whose elements in the leading diagonal are each equal to 1 and all the other elements are zero is a unit matrix. Character is who you are when no one is looking. #3 20060415 10:50:34
Re: Matrix Algebra Formulas"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #4 20060416 17:42:18
Re: Matrix Algebra FormulasEquality of matrices is equal to implies a=1, b=2, c=3, d=4, e=5, f=6, g=7, h=8, and i=9. Singular and nonsingular matrices A square matrix is a singular matrix if A=0. If A≠ 0, then the matrix is nonsingular. A represents the value of the determinant of the matrix. Scalar multiplication of a matrix by a number If then where k is a scalar or a number. Negative of a matrix The negative of a matrix is obtained by multiplying all the elements of the matrix by 1. The negative of is Character is who you are when no one is looking. #5 20060417 00:11:45
Re: Matrix Algebra FormulasAddition and Subtraction of matrices then, Similarly, Multiplication of matrices Two matrices A and B can be multiplied if and only if the number of columns of matrix A is equal to the number of rows of matrix B. If and , then Transpose of a matrix For any given matrix A, the matrix whose rows are columns of matrix A and whose columns are rows of matrix A is called the transpose of matrix A. It is represented by or A'. If matrix A is a m x n matrix, then A' is n x m matrix. If then If asquare matrix and its transpose are equal, then the matrices are symmetric. Properties of Transpose 1. If A and B are two matrices of the same order, then 2. If A and B are compatible for multiplication, then These results can be extended to n matrices. Determinant of a matrix Consider the matrix The determinant of the matrix is given by A=a11(a22*a33a32*a23)a12(a21*a33a31*a23)+a13(a21*a32a31*a22) Let be the matrix obtained by deleting the ith row and jth column of matrix A. The determinant is called a minor of the matrix A. The scalar is called the cofactor of the elelment of the matrix A. Character is who you are when no one is looking. #6 20060417 00:48:19
Re: Matrix Algebra FormulasProducts of Vectors and Matrices Vector Product two vectors containing same amount of entries can be multiplied. some people call this "dot product" or equivalently, using matrix rule of product expression: notice the second vector is placed verticle in matrix rule of product expression. We can think the product as each entry of the former vector(a,b and c) , is scalar multiplied by corresponding entry of the latter vector, and then the 3 product ad, be and cf are added up and give the final result. So does the reverse.(this concept will be applied below next *) Extension to Matrix Vector Product We can add another vector [a_{2} b_{2} c_{2}]under the vector [a b c] and let it do the SAME multiplication to [d e f], and do the SAME summation and the result is placed under previous one for vector[a_{2} b_{2} c_{2}] has been placed under[a b c]: Recall the concept of scaler productsum analyze, we will notice both a and a_{2} are multiplied by scalar d, both b and b_{2} are multiplied by e, as well as both c and c_{2} are multiplied by f, and then corresponding product are added. If we define then the matrix product can be expressed as where col stands for column, A is now a matrix and no longer a vector . we can add many rows [a_{3} b_{3} c_{3}]... [a_{m} b_{m} c_{m}] to the matrix, but still holds. or This is called a matrix multiplied by a vector on the right is equivalent to get its columns(also vectors) combined by entries of the right vector.It's another way to perceive matrice product. Similarly, we have this formula about row combination* Extension to Matrice Product what if a matrix multiplied by a matrix? we can either seperate right matrix(B) into columns of scalars and get a Row of Column combinations or seperate left matrix(A) into rows of scalars and get a Column of Row combinations. Last edited by George,Y (20060420 13:38:19) X'(yXβ)=0 #7 20060420 01:54:15
Re: Matrix Algebra FormulasThis perception is mainly used in vector combination. Still, it's useful in interpret matrices. X'(yXβ)=0 #8 20060424 00:02:09
Re: Matrix Algebra FormulasAdjoint of a square Matrix The adjoint of A is defined to be the transpose of the cofactor matrix of A and is denoted by adj.A. Inverse of a Matrix The Inverse or Reciprocal of a nonSingular Matrix A is denoted by . It can be shown that If A and B are two matrices such that AB=BA=1, then Symmetric and Skewsymmetric Matrices A square matrix is said be symmetric if the (I,j)th element of the matrix is equal to the (j,i)th element. for all values of i and j. A square matrix is said to be skew symmetric if the (i,j)the element is equal to the negative of the (j,i)th element. for all values of i and j. Eamples: Symmetric matrices Skewsymmetric Matrix Conjugate of a Matrix The Matrix obtained from any given Matrix A by replacing its elements by the corresponding conujugate complex numbers is called the conjugate of A and denoted by If then Hermitian and skewHermitian Matrices A square matrix is said to be Hermitian if the (i,j)th element of A is equal to the conjugate complex of the (j,i)th element of A. A square matrix is said to be skewHermitian if the (i,j)th element of A is equal to the negative of the conjugate complexof the (j,i)th element. Example : Hermitian Matrix Example: SkewHermitian Matrix Character is who you are when no one is looking. #9 20060424 01:29:34
Re: Matrix Algebra FormulasProperties of Matrices 12. A (adj A) = (adj A) A = (determinant A)I Adj(AB) = (adjB)(adjA) determinant (adj A) = (determinant A)¹ 13. Character is who you are when no one is looking. #10 20060424 01:46:13
Re: Matrix Algebra FormulasSimultaneous Linear Equations Let and Matrix X is the solution of the given simultaneous equations. Character is who you are when no one is looking. #11 20060426 01:08:23
Re: Matrix Algebra FormulasProperties of Determinants The value of the determinant is unaltered if its rows and columns are interchanged. If two adjacent rows or columns are interchanged, the sign of the determinant changes, but its numerical value remains unaltered. Similarly, If two rows or columns of a determinant are identical, the value of the determinant is zero. If all the elements of one row or one column be multiplied by a nonzero constant k, then the value of the determinant is multiplied by k. If each element of a row or column is expressed as the sum of two numbers, then the determinant can be expressed as a sum of two determinants of the same order. A fetrminant is unaltered in value, by adding to all the lements of any column or any row the same multiple of the corresponding elements of any number of other columns or rows. In any determinant, if the elements of any row or column are multiplied by the cofactors of the corresponsing elements of any other row or column, the sum of the products would be equal to zero. Character is who you are when no one is looking. #12 20060426 02:10:45
Re: Matrix Algebra FormulasCramer's Rule for solving Simultaneous Equations Let the system of equations can be solved by this method. Character is who you are when no one is looking. #14 20060426 14:21:39
Re: Matrix Algebra FormulasGeorge,Y, where i=1, j=2 (the 1st row, 2nd column) The power of (1) takes care of the symbol, hence, IS CORRECT! Character is who you are when no one is looking. #15 20060924 11:09:47
Re: Matrix Algebra Formulaswhat is A inverse Whole Inverse Equal to? Be Happy! 