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#1 2006-03-29 10:03:49

MathsIsFun
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Registered: 2005-01-21
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Matrix Algebra Formulas

Matrix Algebra Formulas


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#2 2006-04-10 17:11:14

ganesh
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Registered: 2005-06-28
Posts: 14,835

Re: Matrix Algebra Formulas

Matrices, Introduction and different types

A matrix is a rectangular array of numbers.

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.

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#3 2006-04-14 12:50:34

Ricky
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Registered: 2005-12-04
Posts: 3,791

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..."

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#4 2006-04-15 19:42:18

ganesh
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Registered: 2005-06-28
Posts: 14,835

Re: Matrix Algebra Formulas

Equality of matrices

Two matrices A and B of the same order are equal when their corresponding elements are equal.


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


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#5 2006-04-16 02:11:45

ganesh
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Registered: 2005-06-28
Posts: 14,835

Re: Matrix Algebra Formulas

Addition and Subtraction of matrices

Matrices can be added or subtracted only if they are of the same order.
Let

and

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*a33-a32*a23)-a12(a21*a33-a31*a23)+a13(a21*a32-a31*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.


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#6 2006-04-16 02:48:19

George,Y
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Registered: 2006-03-12
Posts: 1,306

Re: Matrix Algebra Formulas

Products of Vectors and Matrices

Scalar Product

A vector or a matrix can be multiplied by a scalar k entirely.
For Example

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[sub]2[/sub] b[sub]2[/sub] c[sub]2[/sub]]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[sub]2[/sub] b[sub]2[/sub] c[sub]2[/sub]] has been placed under[a b c]:


Recall the concept of scaler product-sum analyze, we will notice both a and a[sub]2[/sub] are multiplied by scalar d, both b and b[sub]2[/sub] are multiplied by e, as well as both c and c[sub]2[/sub] 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[sub]3[/sub] b[sub]3[/sub] c[sub]3[/sub]]... [a[sub]m[/sub] b[sub]m[/sub] c[sub]m[/sub]] 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 (2006-04-19 15:38:19)


X'(y-Xβ)=0

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#7 2006-04-19 03:54:15

George,Y
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Registered: 2006-03-12
Posts: 1,306

Re: Matrix Algebra Formulas

This perception is mainly used in vector combination. Still, it's useful in interpret matrices.

one application is now you can spot immediately why an identity matrix multiplied by a matrix equals the matrix itself through row selection, and why a matrix multiplied by an identity matrix equals the matrix itself through column selection.


X'(y-Xβ)=0

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#8 2006-04-23 02:02:09

ganesh
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Registered: 2005-06-28
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Re: Matrix Algebra Formulas

Adjoint of a square Matrix

If A is 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 non-Singular 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 Skew-symmetric 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

Skew-symmetric 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 skew-Hermitian 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 skew-Hermitian 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:- Skew-Hermitian Matrix


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#9 2006-04-23 03:29:34

ganesh
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Registered: 2005-06-28
Posts: 14,835

Re: Matrix Algebra Formulas

Properties of Matrices

1. If A and B are two matrices of the same order,
A+B=B+A (Commutative law of addition)

2. If A,B, and C are matrices of the same order,
(A+B)+C=A+(B+C) (Associative law of Addition)

3. If A and B are matrices of the same order and k is a scalar,
k(A+B)=kA+kB

4. A+0=A

5. A +(-A)=0

6. A+B=A+C implies B+C.

7. If A, B, and C are matrices of order mxn, nxp, pxq respectively,
A(BC) = (AB)C (Associative law of multiplication).

8. If A, B, and C are matrices oforder mxn, nxp, pxq respectively, then
A(B+C) = AB + AC (Distributive law)

9. AI=IA=0

10. A0=0

11. If n is a positive integer,

12. A (adj A) = (adj A) A = (determinant A)I

Adj(AB) = (adjB)(adjA)

determinant (adj A) = (determinant A)-¹

13.


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#10 2006-04-23 03:46:13

ganesh
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Posts: 14,835

Re: Matrix Algebra Formulas

Simultaneous Linear Equations

System of Linear equations can be solved with the help of Matrices.

Consider the linear equations



Let

and

Matrix X is the solution of the given simultaneous equations.


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#11 2006-04-25 03:08:23

ganesh
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Registered: 2005-06-28
Posts: 14,835

Re: Matrix Algebra Formulas

Properties of Determinants

Consider the third order determinant

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 non-zero 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.









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#12 2006-04-25 04:10:45

ganesh
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Registered: 2005-06-28
Posts: 14,835

Re: Matrix Algebra Formulas

Cramer's Rule for solving Simultaneous Equations

If



Let





the system of equations can be solved by this method.


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#13 2006-04-25 14:26:02

George,Y
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Registered: 2006-03-12
Posts: 1,306

Re: Matrix Algebra Formulas

should that be

?


X'(y-Xβ)=0

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#14 2006-04-25 16:21:39

ganesh
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Registered: 2005-06-28
Posts: 14,835

Re: Matrix Algebra Formulas

George,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.

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#15 2006-09-23 13:09:47

WiZaRd
Real Member
Registered: 2006-09-22
Posts: 570

Re: Matrix Algebra Formulas

what is A inverse Whole Inverse Equal to?


Be Happy!

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#16 2014-05-23 01:53:52

zetafunc
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Registered: 2014-05-21
Posts: 123

Re: Matrix Algebra Formulas

Determinant of an n x n Matrix

Let A be an n x n matrix with entries (a_ij). Then, the determinant of A is given by

where S_n is the group of permutations of {1, 2, ... , n}.

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#17 2014-05-23 02:43:30

zetafunc
Member
Registered: 2014-05-21
Posts: 123

Re: Matrix Algebra Formulas

Triangular Matrices

Let A be an n x n matrix. Then, A is lower triangular if a_ij = 0 whenever i < j. Similarly, A is upper triangular if a_ij = 0 whenever i > j. The determinant of a triangular matrix is given by the product of the diagonal entries, i.e.

.

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#18 2014-05-23 02:50:18

zetafunc
Member
Registered: 2014-05-21
Posts: 123

Re: Matrix Algebra Formulas

Effects of Row/Column Operations on Determinants

Let the matrix B be obtained from the matrix A by applying the row operation e. Then:

.

These results hold true for the corresponding column operations.

Last edited by zetafunc (2014-05-23 02:54:01)

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