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Definition

A matrix is an array of elements organised in rows and columns The size of on array can be described by row $\times$ columns such as a 2 x 4 matrix $$\left(\begin{array}{cccc}1 & 4 & -1 & 1 \ 2 & 3 & 0 & 2\end{array}\right)$$

A-Level Further Core 1

Identity Matrix

The identity matrix is the matrix $I$ that is all zero apart from having 1's on the leading diagonal. $$ \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ \end{pmatrix} $$ It has the unique property that $AI = IA = A$ for all matrices. This makes it the [[Groups#Flashcards/Flashcards/Axioms of Groups|Identity Element]] of a [[Groups|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 $A$, $B$ can be multiplied if $A$ has dimensions $n \times m$, and $B$ has dimensions $m \times k$. The resulting matrix will have dimensions $n \times k$ (If this condition is met then $A$ is said to be multiplicatively conformable with $B$). 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 $AB = C$, each element $C_{ij}$ is the dot product of the $i$-th row of $A$and the $j$-th column of $B$ For example $$ \left(\begin{array}{ccc} 5 & -1 & 2 \ 8 & 3 & -4 \end{array}\right) \times\left(\begin{array}{cc} 2 & 2 \ 9 & -3 \ 7 & 4 \end{array}\right)=\left(\begin{array}{cc} 15 & 21 \ 15 & -9 \end{array}\right) $$

Minor

A minor of a element $a_{ij}$ is the determinant of the submatrix formed by removing the $i$th row and the $j$th column often denoted as $M_{ij}$

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 $|A|$ or $\det(A)$ If $|A|=0$, $A$ is a singular matrix, however if $|A| \neq 0$, $A$ is non-singular

2x2

The determinant of a $2 \times 2$ matrix $\left(\begin{array}{ll}a & b \ c & d\end{array}\right)$ is a $ad-bc$

3x3

For a $3 \times 3$ the determinant $=A_{11}M_{11} - A_{12}M_{12}+ A_{13}M_{13}$

Transpose

The transpose of a matrix is found by interchanging the rows and the columns. For example, if $A = \begin{pmatrix} 1 & 2 \ 4 & 3 \end{pmatrix}$, $A^{T} = \begin{pmatrix} 1 & 4 \ 2 & 3 \end{pmatrix}$.

Matrix Inverses

The inverse of any non-singular matrix $A^{-1}$ is the matrix such that $AA^{-1} \equiv A^{-1}A \equiv I$ If $A, B$ are non singular matrices then $(AB)^{-1} = B^{-1}A^{-1}$

2x2

$A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, then $A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$

3x3

For a given $3 \times 3$ matrix $A$, $A^{-1} =\displaystyle\frac{1}{|A|}C^{T}$ where $C$ is the matrix of cofactors. To find $C$

  • 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 $$\begin{pmatrix} + & - & + \ - & + & - \ + & - & + \end{pmatrix}​$$

Solving Systems of Linear Equations

If $A\begin{pmatrix} x \ y \ z \end{pmatrix} = v$ then $\begin{pmatrix} x \ y \ z \end{pmatrix} = A^{-1}v$ If A is non-singular then a unique solution for $\begin{pmatrix} x \ y \ z \end{pmatrix}$ can be found for any vector v To solve a given system of equations for x, y, z: $$ \begin{aligned} ax + by + cz& = j \ dx + ey + fz& = k \ gx + hy + iz& = l \ \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} &= \begin{pmatrix} j \ k \ l \end{pmatrix} \ \end{aligned}$$ $$\begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}^{-1} \begin{pmatrix} j \ k \ l \end{pmatrix}$$

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 ![[Linear-Equation-Consistency-1.png]] ![[Linear-Equation-Consistency-2.png]]

A-Level Further Pure 2

Eigenvectors and Eigenvalues

An Eigenvector of Matrix A is a non-zero column [[Vectors|Vector]] $x$ that satisfies $Ax = \lambda x$ where $\lambda$ is a scalar. The value of $\lambda$ is the eigenvalue corresponding to the eigen vector x.

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

Characteristic Equation

$$ \begin{aligned} &Ax = \lambda x = \lambda Ix \ &Ax - \lambda Ix = (A- \lambda I)x = 0\ \Rightarrow&\det(A-\lambda I) = 0 \end{aligned} $$ The solutions of $\lambda$ are the eigenvalues of $A$

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 $0$ they can be use to solved [[Recurrence Relations]] or [[Differential Equations|Coupled Differential Equations]]

To diagonalize a matrix $A$

  • Find eigenvectors of A
  • Form matrix $P$ which consists of these column eigenvectors
  • Find $P^{-1}$
  • The diagonal matrix $D$ is given as $P^{-1}AP$

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

  • Find normalized eigenvectors of $A$
  • Form matrix $P$ with these vectors
  • Find $P^T$
  • $D = P^TAP$ When diagonalizing $A \rightarrow D$, $D$ has the eigen values of A on the leading diagonal *(the eigenvalue in column $n$ corresponds to the eigenvector in column $n$ in $P$)

Powers of Matrices

For any $n \times n$ diagonal matrix $D = \begin{pmatrix} a & 0 \ 0 & d \end{pmatrix}$, $D^k = \begin{pmatrix} a^k & 0 \ 0 & d^k \end{pmatrix}$ To find higher powers of any general matrix $A$ you can use: $$ \begin{aligned} A &= PDP^{-1}\ \Rightarrow A^k &= (PDP^{-1})^k\ &= (PDP^{-1})(PDP^{-1})...(PDP^{-1})\ &=PD(P^{-1}P)D(P^{-1}P)...(P^{-1}P)DP^{-1}\ &= PD^kP^{-1} \end{aligned} $$

Cayley-Hamilton Theorem

All matrices $M$ satisfy their own characteristic equation $$ \begin{aligned} &\det(M-\lambda I) = 0 \ \Rightarrow &a_n{\lambda}^{n} + a_{n-1}{\lambda}^{n-1} + \cdots a_2{\lambda}^2 + a_1{\lambda} + a_0 = 0 \ \Rightarrow &a_n{M}^{n} + a_{n-1}{M}^{n-1} + \cdots a_2{M}^2 + a_1{M} + a_0 I = 0 \end{aligned} $$ This theorem can be used to find various variations of M such as $M^{-1}$ and $M^k$ by multiplying or dividing through by $M$ and using the fact that $I = MM^{-1} \Rightarrow I \div M = M^{-1}$

Flashcards

A-Level Further Core 1

#z_Legacy/Maths/A-Levels/Further-Core/Matrices, #Maths/Topics/Linear-Algebra/Matrices

What the identity matrix? ? The identity matrix is a square matrix where all elements are 0 except for the diagonal from top left down, which contains 1s. It is denoted as $I$

What are the conditions for matrix multiplication and how do you compute $AB$? ? 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 $AB = C$, each element $C_{ij}$ is the dot product of the $i$-th row of $A$and the $j$-th column of $B$ For example $\begin{pmatrix} 5 & -1 & 2 \ 8 & 3 & -4 \end{pmatrix}$$\times$ $\begin{pmatrix} 2 & 2 \ 9 & -3 \ 7 & 4 \end{pmatrix}$=$\begin{pmatrix} 15 & 21 \ 15 & -9 \end{pmatrix}$

How do you calculate the determinant of a 2×2 matrix? ? For a 2×2 matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$, the determinant is $ad - bc$.

What are the properties of an inverse matrix? ? The inverse of any non-singular matrix $A^{-1}$ is the matrix such that $AA^{-1} \equiv A^{-1}A \equiv I$ If $A, B$ are non singular matrices then $(AB)^{-1} = B^{-1}A^{-1}$

How do you find the inverse of a 2×2 matrix? ? $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, then $A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$

How can you solve a system of linear equations using matrices? ? If $A\begin{pmatrix} x \ y \ z \end{pmatrix} = v$, then $\begin{pmatrix} x \ y \ z \end{pmatrix} = A^{-1}v$. Multiply both sides by the inverse of A to find the solution.

What does it mean for a system of linear equations to be consistent? ? A system of linear equations is consistent if there exists at least one set of values that satisfy all equations simultaneously.

How do you compute the determinant of a 3x3 matrix? ? For a $3 \times 3$ the determinant $=A_{11}M_{11} - A_{12}M_{12}+ A_{13}M_{13}$

How do you compute the inverse of a 3x3 matrix? ? For a given $3 \times 3$ matrix $A$, $A^{-1} =\displaystyle\frac{1}{|A|}C^{T}$ where $C$ is the matrix of cofactors. To find $C$

  • 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 $$\begin{pmatrix} + & - & + \ - & + & - \ + & - & + \end{pmatrix}​$$

What are the different ways that a system of linear equations can be consistent/inconsistent? ? 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 ![[Linear-Equation-Consistency-1.png|400]] ![[Linear-Equation-Consistency-2.png|400]]

A-Level Further Pure 2

#Maths/Topics/Linear-Algebra/Matrices, #z_Legacy/Maths/A-Levels/Further-Pure/Matrix-Algebra

How do you diagonalize a matrix $A$? ? To diagonalize matrix $A$:

  1. Find Eigen vectors of $A$
  2. Form matrix P consisting of these Eigen vectors
  3. Find $P^{-1}$
  4. Calculate the diagonal matrix D as $D = P^{-1}AP$ But for an easier calculation you can simply for a diagonal matrix where every element is an Eigen value of $A$ (The eigen value of a specific column should correspond to the eigen vector in that specific column of $P$)

How can you diagonalize a symmetric matrix $A$ more efficiently? ? For a symmetric matrix A (where $A = A^T$):

  1. Find normalized eigen vectors of $A$
  2. Form matrix $P$ with these vectors
  3. Find $P^T$
  4. Calculate $D = P^T AP$ This process is called orthogonal diagonalization.

How can you calculate higher powers of any matrix $A$ using diagonalization? ? To find $A^k$:

  1. Diagonalize $A$ as $A = PDP^{-1}$
  2. Then $A^k = (PDP^{-1})^k = PD^kP^{-1}$

What does the Cayley-Hamilton Theorem state? ? The Cayley-Hamilton Theorem states that any matrix $M$ satisfies its own characteristic equation. If $a\lambda^2 + b\lambda + c = 0$ is the characteristic equation, then $aM^2 + bM + cI = 0$, where $\lambda$ is an eigen value of $A$.

What is an Eigenvector and how do you compute them ? ? An Eigen vector of Matrix $A$ is a non-zero Column Vector $x$ that satisfies the equation $Ax = \lambda x$ where $\lambda$ is a scalar called the eigen value corresponding to the eigen vector $x$. It represents the invariant vector under the linear transform $A$ (although its magnitude may change). The solutions to the characteristic equation of A, given by $det(A-\lambda I) = 0$, are the Eigen values of A.