Keo's Notes

LEGACY Matrices

6 min read

Definition

A matrix is an array of elements organised in rows and columns The size of on array can be described by row columns such as a 2 x 4 matrix

A-Level Further Core 1

Identity Matrix

The identity matrix is the matrix that is all zero apart from having 1’s on the leading diagonal.

It has the unique property that for all matrices. This makes it the Identity Element of a Group of matrices under the operation of multiplication hence the name

Matrix Operations

Addition

To add two matrices simply add the corresponding elements

Multiplication by a Scalar

To multiply a matrix by a scalar simply multiply every individual element by the scalar

Matrix Multiplication

Matrices , can be multiplied if has dimensions , and has dimensions . The resulting matrix will have dimensions (If this condition is met then is said to be multiplicatively conformable with ). Matrix multiplication is associative but not commutative.

To compute a matrix multiplication, take the dot product of each row of the first matrix with each column of the second matrix. Also worded as: For , each element is the dot product of the -th row of and the -th column of For example

Minor

A minor of a element is the determinant of the submatrix formed by removing the th row and the th column often denoted as

Determinant

The determinant is a scalar value associated with that matrix and is the volume enclosed by the Vectors that make up the matrix. It is denoted as or If , is a singular matrix, however if , is non-singular

2x2

The determinant of a matrix is a

3x3

For a the determinant

Transpose

The transpose of a matrix is found by interchanging the rows and the columns. For example, if , .

Matrix Inverses

The inverse of any non-singular matrix is the matrix such that If are non singular matrices then

2x2

, then

3x3

For a given matrix , where is the matrix of cofactors. To find

  • Form the matrix of the minors. This is where each of the nine elements of the matrix is replaced by its minor
  • Change the signs of some elements with alternating signs as shown

Solving Systems of Linear Equations

If then If A is non-singular then a unique solution for can be found for any vector v To solve a given system of equations for x, y, z:

Linear Equation Consistency

A system of linear equations is consistent if at least one set of values that satisfy all equations simultaneously. If the matrix corresponding to a system is non-singular then the system has one solution and is consistent.

However if it is singular then either

  • The system is consistent and has infinitely many solutions
  • It is inconsistent with no solutions

A-Level Further Pure 2

Eigenvectors and Eigenvalues

An Eigenvector of Matrix A is a non-zero column Vector that satisfies where is a scalar. The value of is the eigenvalue corresponding to the eigen vector x.

An eigenvector is an invariant vector under the linear transform . The corresponding eigenvalue can be thought of as the magnitude scale factor for the eigenvector

Characteristic Equation

The solutions of are the eigenvalues of

Reducing Matrices to Diagonal Form

A diagonal matrix is a square matrix such that every element except those which lie on the leading diagonal are they can be use to solved Recurrence Relations or Coupled Differential Equations

To diagonalize a matrix

  • Find eigenvectors of A
  • Form matrix which consists of these column eigenvectors
  • Find
  • The diagonal matrix is given as

If (if A is symmetric about the leading diagonal) then you can find D quicker by orthogonal diagonalization.

  • Find normalized eigenvectors of
  • Form matrix with these vectors
  • Find
  • When diagonalizing , has the eigen values of A on the leading diagonal *(the eigenvalue in column corresponds to the eigenvector in column in )

Powers of Matrices

For any diagonal matrix , To find higher powers of any general matrix you can use:

Cayley-Hamilton Theorem

All matrices satisfy their own characteristic equation

This theorem can be used to find various variations of M such as and by multiplying or dividing through by and using the fact that

Graph View

Backlinks