Linear Algebra - Matrix

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

Example

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

where:

Rows and columns are vectors:

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

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

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:

• scalar-vector multiplication
• vector addition

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

Example

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>

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]

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

Operations

Transpose

Transpose swaps rows and columns. See Dimensional Data Operation - (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>

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 Linear Algebra - Function (Set) 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.

Null Space

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

Nullity

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

• dim Null A

Type

One-column / One-row

A vector can be:

• a one-column matrix called a column vector
• a one-row matrix called a row vector

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})

Diagonal

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

Triangular

Linear Algebra - Triangular Matrix

Column-orthogonal

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

Orthogonal

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

Property

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

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

can be represented as a table

Operations

Multiplication

• Linear Algebra - Matrix Vector (Multiplication)
• Linear Algebra - Matrix Matrix (Multiplication)

Addition/Subtraction

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