Linear Algebra - Matrix

> Linear Algebra

1 - About

The Traditional notion of a matrix is:

  • a two-dimensional array
  • a rectangular table

of known or unknown numbers

One simple role for a matrix: packing together a bunch of columns or rows

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3 - Example

Matrix <math>A_(i,j)</math> of <math>A[i,j]</math>

<math>[email protected]</math> <math>j=\#</math> <math>j=\%</math>
<math>i=a</math> 1 2 3
<math>i=b</math> 4 5 6

where:

Index Type In Matrix A
i rows 2 rows [1,2,3] and [4,5,6]
j columns 3 columns: [1,4], [2,5] and [3,6]

Rows and columns are vectors:

  • Row ’a’ is the vector [1,2,3]
  • Column '%' is the vector [3,6]

4 - Order

The size, or order, of a matrix is given by identifying the number of rows and columns.

The order of matrix A is 2×3

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5 - Function

Matrix as a function: An R x C Matrix over the field F is a function from R x C to F where:

  • R is finite set of unique Row labels: {a, b}
  • C is finite set of unique Column labels: {@, #, %}

As it's a function it can be interpreted as an R x C-vector:

From matrix to function: <math>f (x) = M * x</math> where M is a matrix

If M is an R x C matrix over <math>\mathbb{F}</math> then

  • domain of f is <math>\mathbb{F}^C</math>
  • co-domain of f is <math>\mathbb{F}^R</math>

Example of function:

  • Rotation
  • Translation
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5.1 - Example

5.1.1 - Example 1

Let M be the matrix <math> \begin{array}{r|rrr} & \# & @ & ? \\ \hline a & 1 & 2 & 3 \\ b & 10 & 20 & 30 \end{array} </math> and define <math>f ({\bf x}) = M * {\bf x}</math> then:

  • domain of f is <math>\mathbb{F}^{\#,@,?}</math>
  • co-domain of f is <math>\mathbb{F}^{a,b}</math>

f maps <math> \begin{array}{rrr} \# & @ & ? \\ \hline 2 & 2 & -2 \\ \end{array} </math> to <math> \begin{array}{rr} a & b \\ \hline 0 & 0 \\ \end{array} </math>

5.1.2 - Example 2

Define <math> f({\bf x}) = \begin{bmatrix} 1 & 2 & 3 10 & 20 & 30 \end{bmatrix} * {\bf x} </math>:

  • domain of f is <math>\mathbb{R}^3</math>
  • co-domain of f is <math>\mathbb{R}^2</math>

f maps [2, 2,-2] to [0,0]

6 - Dimension

For the following function

  • <math>f : \mathbb{F}^A \rightarrow \mathbb{F}^B</math>
  • also noted as <math>f : \mathbb{F}^n \rightarrow \mathbb{F}^m</math>
  • <math>f ({\bf x}) = M * {\bf x}</math>

we know:

  • Since the domain is <math>\mathbb{F}^A</math>, that the input x is an A-vector.
  • Since the co-domain is <math>\mathbb{F}^B</math>, that the output is a B-vector
  • then that M must be a B x A matrix

7 - Operations

7.1 - Transpose

Transpose swaps rows and columns. See Data Visualisation - (Pivot|Transpose|Cross-tab|Matrix)

<MATH> \begin{bmatrix} \begin{array}{rrr} 4 & 1 & -3 \\ 2 & 2 & -2 \end{array} \end{bmatrix}^T = \begin{bmatrix} \begin{array}{rr} 4 & 2 \\ 1 & 2 \\ -3 & -2 \end{array} \end{bmatrix} </MATH>

<MATH> \begin{bmatrix} \begin{array}{r} 4 \\ 1 \\ -3 \end{array} \end{bmatrix}^T = \begin{bmatrix} \begin{array}{rrr} 4 & 1 & -3 \end{array} \end{bmatrix} </MATH>

<MATH> \begin{bmatrix} \begin{array}{rr} 4 & 2 \\ 1 & 2 \\ -3 & -2 \end{array} \end{bmatrix}^T = \begin{bmatrix} \begin{array}{rrr} 4 & 1 & -3 \\ 2 & 2 & -2 \\ \end{array} \end{bmatrix} </MATH>

7.2 - Inverting

Only square matrices can be inverted, and square matrices are not guaranteed to have an inverse. If the inverse exists, then multiplying a matrix by its inverse will produce the identity matrix.

Theorem: The transpose of an invertible matrix is invertible.

If <math>A</math> has an inverse <math>A^{-1}</math> then <math>AA^{-1}</math> is identity matrix

Converse: If BA is identity matrix then A and B are inverses? Not always true.

Theorem: Suppose A and B are square matrices such that BA is an identity matrix 1. Then A and B are inverses of each other.

Matrices A and B are inverses of each other if and only if both AB and BA are identity matrices.

<MATH> A B = A A^{-1} = I_n </MATH>

A invertible matrix has an inverse.

Corollary

Let A be an R x C matrix. Then A is invertible if and only if |R| = |C| and the columns of A are linearly independent.

Proof: Let <math>\mathbb{F}</math> be the field. Define <math>f : \mathbb{F}^C \mapsto \mathbb{F}^R</math> by <math>f (x) = Ax</math> Then A is an invertible matrix if and only if f is an invertible function.

The function f is invertible

  • iff dim Ker f = 0 and dim <math>\mathbb{F}^C</math> = dim <math>\mathbb{F}^R</math>
  • iff nullity A = 0 and |C| = |R|.

nullity A = 0

  • iff dim Null A = 0
  • iff Null A = {0}
  • iff the only vector x such that Ax = 0 is x = 0
  • iff the columns of A are linearly independent

Example:

  • <math>\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix}</math> is not square and so cannot be invertible.
  • <math>\begin{bmatrix}1 & 2 \\ 3 & 4 \end{bmatrix}</math> is square and its columns are linearly independent so it is invertible.
  • <math>\begin{bmatrix}\begin{array}{r|r|r}1 & 1 & 2 \\ 2 & 1 & 3 \\ 3 & 1 & 4 \end{array}\end{bmatrix}</math> is square but its columns are not linearly independent so it is not invertible.

8 - Null Space

Null space of a matrix A (Written Null A) is: <MATH> \{u : A * u = 0\} </MATH>

9 - Nullity

The nullity of matrix A is the dimension of the Null Space written:

  • dim Null A

10 - Type

10.1 - One-column / One-row

A vector can be:

10.2 - Identity

D x D identity matrix is the matrix <math>1_D</math> such that <math>1_D [k, k] = 1 \text{ for all } k \in D</math> and zero elsewhere.

Often letter I (for “identity”) is used instead of 1

<MATH> \mathbf{I_n} = \begin{bmatrix} 1 & 0 & 0 & \dots & 0 \\\ 0 & 1 & 0 & \dots & 0 \\\ 0 & 0 & 1 & \dots & 0 \\\ \vdots & \vdots & \vdots & \ddots & \vdots \\\ 0 & 0 & 0 & \dots & 1 \end{bmatrix} </MATH>

def identity(D): return Mat((D,D), {(k,k):1 for k in D})

10.3 - Diagonal

<math> \begin{bmatrix} d_1 & & \\ & \ddots & \\ & & d_3 \end{bmatrix} </math>

Let <math>d_1, \dots , d_n</math> be real numbers. Let <math>f : \mathbb{R}^n \rightarrow \mathbb{R}^n </math> be the function such that <math>f ([x_1, \dots , x_n]) = [d_1*x_1, \dots , d_n*x_n]</math>.

For a domain D, a D x D matrix M is a diagonal matrix if M[r , c] = 0 for every pair <math>r, c \in D</math> such that <math>r \neq c</math>.

A matrix is called a diagonal matrix when the only entries allowed to be nonzero form a diagonal

10.4 - Triangular

10.5 - Column-orthogonal

If columns of a matrix are orthonormal then it's a column-orthogonal matrix (should be called orthonormal ….)

10.6 - Orthogonal

If a matrix is square and column-orthogonal, it's an orthogonal matrix

11 - Property

11.1 - Rank

For a matrix M:

  • the row rank of M is the rank of its rows,
  • and the column rank of M is the rank of its columns.

Equivalently, the row rank of M is the dimension of Row M, and the column rank of M is the dimension of Col M.

Consider the matrix <MATH> M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 4 & 0 \end{bmatrix} </MATH> where:

  • The rows are the vectors: [1, 0, 0], [0, 2, 0], [2, 4, 0]. The set of these vectors has rank two, so the row rank of M is two.
  • The columns of M are [1, 0, 2], [0, 2, 4], and [0, 0, 0]. Since the third vector is the zero vector, it is not needed for spanning the column space. Since each of the first two vectors has a nonzero where the other has a zero, these two are linearly independent, so the column rank is two.

Rank Theorem: For every matrix M, row rank equals column rank.

Lemma: For any matrix A, row rank of A  column rank of A

11.2 - Sparse

A sparse matrix has many positions with a value of zero.

Systems designed to efficiently support sparse matrices look a lot like databases: They represent each cell as a record (i,j,value).

The benefit is that you only need one record for every non-zero element of a matrix.

For example, the matrix

0 2 -1
1 0 0
0 0 -3
0 0 0

can be represented as a table

row # (i) column # (j) value
0 1 2
0 2 -1
1 0 1
2 2 -3

12 - Operations

12.1 - Multiplication

12.2 - Addition/Subtraction

Two matrices may be added or subtracted only if they are of the same order.

linear_algebra/matrix.txt · Last modified: 2017/09/14 13:24 by gerardnico