Using LaTeX with WordPress

Updated 2017 November 17th

WordPress now supports LaTeX. To display LaTeX, just surround the code with "$latex" and "$", without the quotations. In this post, I show some formulae from this tutorial on SVD. I've shown the code for each formula; you can also mouseover the formula to see the code.

Displaying a vector

\vec{x} = \begin{pmatrix}8\\6\\7\\5\\3\end{pmatrix}

\vec{x} = \begin{pmatrix}8\\6\\7\\5\\3\end{pmatrix}

Displaying a matrix

\begin{bmatrix} 17 & 18 & 5 & 5 & 45 & 1 \\ 42 & 28 & 30 & 15 & 115 & 3 \\ 10 & 10 & 10 & 21 & 51 & 2 \\ 28 & 5 & 65 & 39 & 132 & 5 \\ 24 & 26 & 45 & 21 & 116 & 4 \end{bmatrix}

\begin{bmatrix} 17 & 18 & 5 & 5 & 45 & 1 \\ 42 & 28 & 30 & 15 & 115 & 3 \\ 10 & 10 & 10 & 21 & 51 & 2 \\ 28 & 5 & 65 & 39 & 132 & 5 \\ 24 & 26 & 45 & 21 & 116 & 4 \end{bmatrix}

A = \begin{bmatrix} a_{11} & \cdots & a_{1j} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{i1} & \cdots & a_{ij} & \cdots & a_{in} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mj} & \cdots & a_{mn} \end{bmatrix}

A = \begin{bmatrix} a_{11} & \cdots & a_{1j} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{i1} & \cdots & a_{ij} & \cdots & a_{in} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mj} & \cdots & a_{mn} \end{bmatrix}

Above is an m \times n matrix, where a_{ij} are elements of the matrix A , i = the i_{th} row and j = the j_{th} column. The sequence of numbers:

A_{(i)} = (a_{i1}, \dotsc, a_{in})

A_{(i)} = (a_{i1}, \dotsc, a_{in})

is the i_{th} row of A, and the sequence of numbers

A^{(j)} = (a_{1j}, \dotsc, a_{mj})

A^{(j)} = (a_{1j}, \dotsc, a_{mj})

is the j_{th} column of A.

Calculating vector length

The length of a vector is found by squaring each component, summing them, and taking the square root of the sum. If \vec{v} is a vector, its length is {\lvert}\vec{v}{\rvert} .

{\lvert}\vec{v}{\rvert} = \sqrt{ \sum_{i=1}^{n}{x_i^2} }

{\lvert}\vec{v}{\rvert} = \sqrt{ \sum_{i=1}^{n}{x_i^2} }

Vector addition

A = [a_{1}, a_{2}, \dotsc, a_{n}]

A = [a_{1}, a_{2}, \dotsc, a_{n}]

B = [b_{1}, b_{2}, \dotsc, b_{n}]

B = [b_{1}, b_{2}, \dotsc, b_{n}]

A + B = [a_{1} + b_{1}, a_{2} + b_{2}, \dotsc, a_{n} + b_{n}]

A + B = [a_{1} + b_{1}, a_{2} + b_{2}, \dotsc, a_{n} + b_{n}]

Scalar multiplication

Multiplying a scalar (real number) to a vector means multiplying every component by that real number to yield a new vector.

\vec{v} = [3, 6, 8, 4] \times 1.5 = [4.5, 9, 12, 6]

\vec{v} = [3, 6, 8, 4] \times 1.5 = [4.5, 9, 12, 6]

Inner product

The inner product of two vectors (also called the dot product or scalar product) defines multiplication of vectors. The inner product of two vectors is denoted (\vec{v_{1}}, \vec{v_{2}}) or \vec{v_{1}} \cdot \vec{v_{2}} .

(\vec{x}, \vec{y}) = \vec{x} \cdot \vec{y} = \sum_{i=1}^{n}{x_{i}y_{i}}

(\vec{x}, \vec{y}) = \vec{x} \cdot \vec{y} = \sum_{i=1}^{n}{x_{i}y_{i}}

For example, if \vec{x} = [1, 6, 7, 4] and \vec{y} = [3, 2, 8, 3] , then

\vec{x} \cdot \vec{y} = 1(3) + 6(2) + 7(8) + 3(4) = 83

\vec{x} \cdot \vec{y} = 1(3) + 6(2) + 7(8) + 3(4) = 83

Orthogonality

Two vectors are orthogonal to each other if their inner product equals zero. For example, the vectors [2, 1, -2, 4] and [3, -6, 4, 2] are orthogonal because

[2, 1, -2, 4] \cdot [3, -6, 4, 2] = 2(3) + 1(-6) - 2(4) + 4(2) = 0

[2, 1, -2, 4] \cdot [3, -6, 4, 2] = 2(3) + 1(-6) - 2(4) + 4(2) = 0

Normal vector

A normal vector (or unit vector) is a vector of length 1. If you divide each component by its vector length, you get the normal/unit vector.

If \vec{v} = [2, 4, 1, 2] , then

{\lvert}\vec{v}{\rvert} = \sqrt{2^2 + 4^2 + 1^2 + 2^2} = \sqrt{25} = 5

{\lvert}\vec{v}{\rvert} = \sqrt{2^2 + 4^2 + 1^2 + 2^2} = \sqrt{25} = 5

Then \vec{u} = [2/5, 4/5, 1/5, 1/5] is a normal vector because

{\lvert}\vec{u}{\rvert} = \sqrt{(2/5)^2 + (4/5)^2 + (1/5)^2 + (2/5)^2} = \sqrt{25/25} = 1

{\lvert}\vec{u}{\rvert} = \sqrt{(2/5)^2 + (4/5)^2 + (1/5)^2 + (2/5)^2} = \sqrt{25/25} = 1

Orthonormal vectors

Vectors of unit length that are orthogonal to each other are said to be orthonormal. For example

\vec{u} = [2/5, 1/5, -2/5, 4/5]

\vec{u} = [2/5, 1/5, -2/5, 4/5]

and

\vec{v} = [3 / \sqrt{65}, -6 / \sqrt{65}, 4 / \sqrt{65}, 2 / \sqrt{65}]

\vec{v} = [3 / \sqrt{65}, -6 / \sqrt{65}, 4 / \sqrt{65}, 2 / \sqrt{65}]

are orthonormal because

{\lvert}\vec{u}\rvert = \sqrt{(2/5)^2 + (1/5)^2 + (-2/5)^2 + (4/5)^2} = 1

{\lvert}\vec{u}\rvert = \sqrt{(2/5)^2 + (1/5)^2 + (-2/5)^2 + (4/5)^2} = 1

{\lvert}\vec{v}\rvert = \sqrt{(3 / \sqrt{65})^2 + (-6 / \sqrt{65})^2 + (4 / \sqrt{65})^2 + (2 / \sqrt{65})^2} = 1

{\lvert}\vec{v}\rvert = \sqrt{(3 / \sqrt{65})^2 + (-6 / \sqrt{65})^2 + (4 / \sqrt{65})^2 + (2 / \sqrt{65})^2} = 1

\vec{u} \cdot \vec{v} = \frac{6}{5\sqrt{65}} - \frac{6}{5\sqrt{65}} - \frac{8}{5\sqrt{65}} + \frac{8}{5\sqrt{65}} = 0

\vec{u} \cdot \vec{v} = \frac{6}{5\sqrt{65}} - \frac{6}{5\sqrt{65}} - \frac{8}{5\sqrt{65}} + \frac{8}{5\sqrt{65}} = 0

Square matrix

A matrix is square when m = n . To designate the size of a square matrix with n rows and columns, it is called n-square. A 3-square matrix

A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

Transpose

The transpose of matrix A is A^T

A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}

A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}

A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}

A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}

Matrix multiplication

It is possible to multiply two matrices only when the number of rows of the first matrix matches the number of columns of the second matrix.

AB = \begin{bmatrix} 2 & 1 & 4 \\ 1 & 5 & 2 \end{bmatrix} \begin{bmatrix} 3 & 2 \\ -1 & 4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 9 & 16 \\ 0 & 26 \end{bmatrix}

Here's the working

AB = \begin{bmatrix} 2 & 1 & 4 \\ 1 & 5 & 2 \end{bmatrix} \begin{bmatrix} 3 & 2 \\ -1 & 4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 9 & 16 \\ 0 & 26 \end{bmatrix}

ab_{11} = \begin{bmatrix} 2 & 1 & 4 \end{bmatrix} \begin{bmatrix} 3 \\ -1 \\ 1 \end{bmatrix} = 2(3) + 1(-1) + 4(1) = 9

ab_{12} = \begin{bmatrix} 2 & 1 & 4 \end{bmatrix} \begin{bmatrix} 2 \\ 4 \\ 2 \end{bmatrix} = 2(2) + 1(4) + 4(2) = 16

ab_{21} = \begin{bmatrix} 1 & 5 & 2 \end{bmatrix} \begin{bmatrix} 3 \\ -1 \\ 1 \end{bmatrix} = 1(3) + 5(-1) + 2(1) = 0

ab_{22} = \begin{bmatrix} 1 & 5 & 2 \end{bmatrix} \begin{bmatrix} 2 \\ 4 \\ 2 \end{bmatrix} = 1(2) + 5(4) + 2(2) = 26

Identity matrix

The identity matrix (I ) is a square matrix where all components are 0 expect for components on the diagonal, which are equal to 1. If you multiple a matrix with an identity matrix, you end up with the matrix.

AI = \begin{bmatrix}2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix}2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix}

AI = \begin{bmatrix}2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix}2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix}

Orthogonal matrix

Matrix A is orthogonal if AA^T = A^TA = I . Matrix A is a symmetric matrix since it is equal to its transpose.

A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & -4/5 \\ 0 & 4/5 & 3/5 \end{bmatrix}

A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & -4/5 \\ 0 & 4/5 & 3/5 \end{bmatrix}

is orthogonal because

AA^T = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & -4/5 \\ 0 & 4/5 & 3/5 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & 4/5 \\ 0 & -4/5 & 3/5 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

AA^T = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & -4/5 \\ 0 & 4/5 & 3/5 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & 4/5 \\ 0 & -4/5 & 3/5 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Diagonal matrix

A diagonal matrix A is a square matrix where all the entries a_{ij} are 0 when i \neq j .

A = \begin{bmatrix} a_{11} & 0 & 0 & 0 \\ 0 & a_{22} & 0 & 0 \\ 0 & 0 & a_{33} & 0 \\ 0 & 0 & 0 & a_{mm} \end{bmatrix}

A = \begin{bmatrix} a_{11} & 0 & 0 & 0 \\ 0 & a_{22} & 0 & 0 \\ 0 & 0 & a_{33} & 0 \\ 0 & 0 & 0 & a_{mm} \end{bmatrix}

The trace of an n \times n square matrix is defined to be the sum of the elements on the main diagonal.

tr(A) = \sum^{n}_{i=1} a_{ii} = a_{11} + a_{22} + \dots + a_{nn}

tr(A) = \sum^{n}_{i=1} a_{ii} = a_{11} + a_{22} + \dots + a_{nn}

Determinant

A determinant is a function of a square matrix that reduces it to a single number. The determinant of a matrix A is denoted {\lvert}A\rvert or det(A) . If A consists of one element a , then {\lvert}A\rvert = a . If A is a 2 x 2 matrix, then

{\lvert}A\rvert = \left| \begin{array}{cc} a & b \\ c & d \end{array} \right| = ad - bc

{\lvert}A\rvert = \left| \begin{array}{cc} a & b \\ c & d \end{array} \right| = ad - bc

The determinant of

A = \begin{bmatrix} 4 & 1 \\ 1 & 2 \end{bmatrix}

A = \begin{bmatrix} 4 & 1 \\ 1 & 2 \end{bmatrix}

is

{\lvert}A\rvert = \left| \begin{array}{cc} 4 & 1 \\ 1 & 2 \end{array} \right| = 4(2) - 1(1) = 7

{\lvert}A\rvert = \left| \begin{array}{cc} 4 & 1 \\ 1 & 2 \end{array} \right| = 4(2) - 1(1) = 7

When the determinant of a matrix is 0, the inverse of the matrix does not exist.

Eigenvectors and eigenvalues

An eigenvector is a nonzero vector that satisfies the equation

A\vec{v} = \lambda\vec{v}

A\vec{v} = \lambda\vec{v}

where A is a square matrix, \lambda is a scalar, and \vec{v} is the eigenvector. \lambda is called an eigenvalue.

Useful links

LaTeX – Multiline equations, systems and matrices
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