When stating a matrix the numer of rows is stated first and the number of columns is stated second.
e.g. a 2x3 matrix has 2 rows and 3 columns
A matrix is typically denoted with A, the elemments are donated with the same lower case letter
e.g. a11 for the element in row 1. column 1.
For a m x n matrix, if n = m then A is a square matrix of order n, the elements a11, a22, ... , ann form the main diagonal.
Matrix addition and subtraction
R = A + B
>A and B must be the same size
>R is the same size.
> in R for all the elements rij = aij + bij
for the subtraction of 2 matricies A and B, R = A - B
>Subtraction has the same rules as addition with 1 exception:
In R for all the elements rj = aij - bij
Matrix A = B if they are of the same size and the correspoding elements are the same
i.e. aij - b ij for all i,j
R = k*A, each elemetns in A is multiplies by a scalar, each rij = K*aij
R = A * B
>Number iof columns in A must equal number of rows in B
>R has number of rows as A and the number of coloums in B
e.g. A 2x3 matrix, B 3x4 matrix A*B 2x3 matrix.
The matrix found by:
Multiply each row by each column then find the sum, and place this sum in the element rij of R
Matrix multiplication 2
Matrix multiplication properties
> if A*B is ok then B*A may not be possible, A*B ≠ B*A
> Commutative - 2 matricies A, B are said to cummute if and only if A*B = B*A
> Associative - matrix multiplication is associative; A * (B * C) = (A * B) * C = A * B * C
> Distributive - matrix multiplication is distrutive; A * (B + C) = (A * B) + (A * C)
A transposed is AT
Here the matrix is flipped: rows become columns and columns become rows.
> If R = AT then rij = aij.
> If A is size mxn then AT is size nxm
Symmetric if A = AT and skew-symmetric if A =-AT
New location of object TT = R(Ө) * T where R(Ө) is the rotation matrix:
To translate, simply add the value to be translated by to the matrix
Translation and roation
multiply by rotation matrix then add the translation matrix
Let A be an n x n matrix. If there exists and n x n matrix B such that AB = I = AB we can say A is invertible and call B the matrix inverse of A.
The inverse of matrix A is denoted A-1
A-1 is defined as:
A matrix has no inverse if its determinant is 0.
If det(A) ≠ 0, the matrix is invertible – non-singular
If det(A) = 0, then A-1 does not exist - singular
Matrix determinants, Adjoints, cofactor and minors
The determinant of A is written as det(A) or |A|
The adjoint of A is denoted Adj(A)
Adj(A) is the transpose of the matrix of the cofactors of A.
If Cij is the cofactor of aji then Adj(A) = [Cji] = [Cij]T
Each element Aij of a matrix A has a cofactor Cij = -1(i+j)*Mij, Mij is the determinant of the matrix which is left when row i and column j are removed from A.
Matrix determinant of 2x2 and 3x3
The determinant of an nxn matrix is the sum of each element in its top row mulitplies by that element's cofactor.
|A| = (a11 * a22) - (a12 * a21)
Matrix adjoint 2x2 and 3x3
Adj(A) is the transpose of the matrix of cofactors of A
Find the matrix of cofactors and then transpose (swap columns and rows)
Properties of Determinants
Let A be an n x n matrix and c be scalar. Then,
Det(cA) = cn det(A)
Let A and B be square matricies of the same size. then,
det(AB) = det(A)det(B)
This can be extended to more matricies, provided they are all of the same size:
det(ABCD...) = det(A)det(B)det(C)det(D)....
Gaussian Elimination and argumented matrix
For a linear equation in matrix form Ax = b, the first step in gaussian elimination is to form the argumented matrix: Ã = [A|b]
There are 3 elementary operations:
- interchange any 2 rows
- multiply (or divide) any row by a non-zero scalar
- Add ( or subtract) a multiple of one row to (or from) another row
When the matrix is in echelon form, gaussian elimination is complete
A matrix is in echelon form if:
- Any rows consisting entirely of zeros are the last rows of the matrix
- As you move down the rows of the matrix, the leading entry (the leftmost non-zero entry) moves progressively to the right.
- All entries in a column below a leading entry are zero
Gauss Jordon method for finding the inverse
First write the argumented matrix in the form [A I]:
Perform elemntary row operations until the matrix is of the form [I B], that is the left half of the matrix is the identity.
Then B, the right half of the matrix will be the inverse of A
A-1 = B
An alternative to Gaussian elimination.
Solutions to a linear system A * x = b, where A is an n x n matrix:
x1 = D1/D
x2 = D2/D
xn = Dn/D
where Dk is det|A|, Det <> 0 and Dk is the determinant of the matrix formed by taking A and replacing its kth column with b.
Impracticable in large matricies.
For a 2x2 matrix multiplied by a 2x1 matrix = 2x1 matrix, to find D1 and D2:
Replace the 1st column of A with b and find the determinant. Then replace the 2nd column of A with b and find the determinant.
Number of solutions
2 parallel lines, after gaussian elimination the matrix final line is 0 = a number
2 lines overlap, after gaussian elimination, the matrix final line is 0 = 0
Number of solutions 2
The rank of a matrix can be used to determine the number of solutions to the equation Ax = b
> the number of non zero rows in the argumented matrix when in row echelon form.
> by determinant, rank of m x n matrix is largest square submatrix whose det ≠ 0. A submatrix of A is a matrix of A minus some rows or columns
Rank and number of solutions
Infinite number of solutions
Det ≠ 0, Rank(Ã) < number of rows
Det ≠ 0, Rank(Ã) > Ranks(A)
Det ≠ 0, Rank(Ã) = Ranks(A)
Fundamental Theorem of linear systems
If system defined by m-row matrix equation A x = b
The system has solutions only if Rank(A) =Rank(Ã)
- if Rank(A) = m, there is 1 solution
- if Rank(A) < m, there is an infinite number of solutions
Properties of Rank, homogeneous systems and cramer
Properties of Rank
- The Rank of A is 0 only if A is the zero matrix
- Rank(A) = Rank(AT)
- Elementary row operations don't effect the rank of a matrix.
- If system defined by Ax = 0 (b=0), system is homogenous.
- A homogenous system always has at least 1 solution, that is a trivial solution, x1 = x2 = ... xn = 0
- A non-trivial solution exists if Rank(A) < m
For a homogeneous system:
- If det(A)≠0 the only solution is x = 0 (non-trivial)
- If det(A) = 0, the system has infinite number of non trivial solutions
Eigenvalues and Eigenvectors
The eigenvalue equation:
Ax = λx
let A be a given n x n matrix. if we can find the non-trivail solution, an n x 1 column vector x and a scalar λ such that Ax = λx then we call λ the eigenvalue and x the associated eignevector.
Here A is nxn matrix, x an nx1 matrix and λ a scalar.
Each (of n) scalar λ satisfying Ax = λx is an eigenvalues of A.
For each λ, an x which satisfies Ax = λx is an eigenvector of A (sometimes labelled Ʌ)
Eigenvalue and Eigenvector process
To solve for the eigenvalues, λI and the corresponding eigenvectors xi, of an nxn matrix A:
- multiply an nxn identity matrix by the scalar
- subtract the identity matrix multiple from the matrix A
- Find the determinant of the matrix and the difference
- Solve for the values of λ that satisfy the equation det(A-λI)x = 0
- Solve the corresponding vector for each λ
Eigenvalue and Eigenvector 2x2
1. Find |A - λ*I|, then find the characteristic polynomial and factories to find λ1 and λ2.
2. For each λ (where x is the eigenvector), find the eigenvector, solve (A - λI)x = 0
- for each λi, substitute into A-λI matrix to create new matrix B
- find the argumented matrix of B
- Use Gaussian elimination to get into Echelon form, to get 2 equations.
- Pick 2 values to sibstitute into the equations, e.g. x1 = 1, x1=2....
Eigenvalue and Eigenvector 3x3
1. find |A-λI| and hence find the characteristic equation
2. solve the characterist equation to find the eigenvalues, this can be one using the integer root theorem:
- look at the constant term and find all the roots. The roots which when substituted into the equation makes LHS = RHS, is a root.
- Hence a factor has been found, divide the equation by this factor.
- Then factorise the last part of the equation to find the next 2 eigenvalues
3. For (A-λI)x=0, find the echelon form matrix of A-λI
4. for the 3 equations find the values for x1, x2 and x3 by giving 1xn a value.
e.g. x1 could = 1
Complex and Repeated Eigenvalues
The process for finding complex and repeated eigenvalues is the same as finding real eigenvalues.