Fundamental Mathematics/Matrix

Matrix

In mathematics, a matrix (plural matrices) is a rectangular array

The individual items in an m × n matrix A, often denoted by a_i,j, where max i = m and max j = n, are called its elements or entries. Provided that they have the same size (each matrix has the same number of rows and the same number of columns as the other).

Two matrices can be added or subtracted element by element (see Conformable matrix). The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second (i.e., the inner dimensions are the same, n for A_m,n × B_n,p). Any matrix can be multiplied element-wise by a scalar from its associated field.

A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three-dimensional space is a linear transformation, which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.

Types of Matrices

Common types of matrices that we encounter in finite elements are:

a row vector that has one row and $n$ columns.

\mathbf {v} ={\begin{bmatrix}v_{1}&v_{2}&v_{3}&\dots &v_{n}\end{bmatrix}}

a column vector that has $n$ rows and one column.

\mathbf {v} ={\begin{bmatrix}v_{1}\\v_{2}\\v_{3}\\\vdots \\v_{n}\end{bmatrix}}

a square matrix that has an equal number of rows and columns.
a diagonal matrix which is a square matrix with only the

diagonal elements ( $a_{ii}$ ) nonzero.

\mathbf {A} ={\begin{bmatrix}a_{11}&0&0&\dots &0\\0&a_{22}&0&\dots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\dots &a_{nn}\end{bmatrix}}~.

the identity matrix ( $\mathbf {I}$ ) which is a diagonal matrix and

with each of its nonzero elements ( $a_{ii}$ ) equal to 1.

\mathbf {A} ={\begin{bmatrix}1&0&0&\dots &0\\0&1&0&\dots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\dots &1\end{bmatrix}}~.

a symmetric matrix which is a square matrix with elements

such that $a_{ij}=a_{ji}$ .

\mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&a_{13}&\dots &a_{1n}\\a_{12}&a_{22}&a_{23}&\dots &a_{2n}\\a_{13}&a_{23}&a_{33}&\dots &a_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\a_{1n}&a_{2n}&a_{3n}&\dots &a_{nn}\end{bmatrix}}~.

a skew-symmetric matrix which is a square matrix with elements

such that $a_{ij}=-a_{ji}$ .

\mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&a_{13}&\dots &a_{1n}\\-a_{12}&a_{22}&a_{23}&\dots &a_{2n}\\-a_{13}&-a_{23}&a_{33}&\dots &a_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\-a_{1n}&-a_{2n}&-a_{3n}&\dots &a_{nn}\end{bmatrix}}~.

Note that the diagonal elements of a skew-symmetric matrix have to be zero: $a_{ii}=-a_{ii}\Rightarrow a_{ii}=0$ .

Matrix Operations

Determinant of a matrix

The determinant of a matrix is defined only for square matrices.

For a $2\times 2$ matrix $\mathbf {A}$ , we have

\mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}\implies \det(\mathbf {A} )={\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}=a_{11}a_{22}-a_{12}a_{21}~.

For a $n\times n$ matrix, the determinant is calculated by expanding into minors as

{\begin{aligned}&\det(\mathbf {A} )={\begin{vmatrix}a_{11}&a_{12}&a_{13}&\dots &a_{1n}\\a_{21}&a_{22}&a_{23}&\dots &a_{2n}\\a_{31}&a_{32}&a_{33}&\dots &a_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&a_{n3}&\dots &a_{nn}\end{vmatrix}}\\&=a_{11}{\begin{vmatrix}a_{22}&a_{23}&\dots &a_{2n}\\a_{32}&a_{33}&\dots &a_{3n}\\\vdots &\vdots &\ddots &\vdots \\a_{n2}&a_{n3}&\dots &a_{nn}\end{vmatrix}}-a_{12}{\begin{vmatrix}a_{21}&a_{23}&\dots &a_{2n}\\a_{31}&a_{33}&\dots &a_{3n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n3}&\dots &a_{nn}\end{vmatrix}}+\dots \pm a_{1n}{\begin{vmatrix}a_{21}&a_{22}&\dots &a_{2(n-1)}\\a_{31}&a_{32}&\dots &a_{3(n-1)}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\dots &a_{n(n-1)}\end{vmatrix}}\end{aligned}}

In short, the determinant of a matrix $\mathbf {A}$ has the value

{\det(\mathbf {A} )=\sum _{j=1}^{n}(-1)^{1+j}a_{1j}M_{1j}}

where $M_{ij}$ is the determinant of the submatrix of $\mathbf {A}$ formed by eliminating row $i$ and column $j$ from $\mathbf {A}$ .

Some useful identities involving the determinant are given below.

If $\mathbf {A}$ is a $n\times n$ matrix, then

\det(\mathbf {A} )=\det(\mathbf {A} ^{T})~.

If $\lambda$ is a constant and $\mathbf {A}$ is a $n\times n$ matrix, then

\det(\lambda \mathbf {A} )=\lambda ^{n}\det(\mathbf {A} )\implies \det(-\mathbf {A} )=(-1)^{n}\det(\mathbf {A} )~.

If $\mathbf {A}$ and $\mathbf {B}$ are two $n\times n$ matrices, then

\det(\mathbf {A} \mathbf {B} )=\det(\mathbf {A} )\det(\mathbf {B} )~.

If you think you understand determinants, take the quiz.

Matrix addition

Let $\mathbf {A}$ and $\mathbf {B}$ be two $m\times n$ matrices with components $a_{ij}$ and $b_{ij}$ , respectively. Then

\mathbf {C} =\mathbf {A} +\mathbf {B} \implies c_{ij}=a_{ij}+b_{ij}

Matrix Multiplication

Multiplication by a scalar

Let $\mathbf {A}$ be a $m\times n$ matrix with components $a_{ij}$ and let $\lambda$ be a scalar quantity. Then,

\mathbf {C} =\lambda \mathbf {A} \implies c_{ij}=\lambda a_{ij}

Multiplication of matrices

Let $\mathbf {A}$ be a $m\times n$ matrix with components $a_{ij}$ . Let $\mathbf {B}$ be a $p\times q$ matrix with components $b_{ij}$ .

The product $\mathbf {C} =\mathbf {A} \mathbf {B}$ is defined only if $n=p$ . The matrix $\mathbf {C}$ is a $m\times q$ matrix with components $c_{ij}$ . Thus,

\mathbf {C} =\mathbf {A} \mathbf {B} \implies c_{ij}=\sum _{k=1}^{n}a_{ik}b_{kj}

Similarly, the product $\mathbf {D} =\mathbf {B} \mathbf {A}$ is defined only if $q=m$ . The matrix $\mathbf {D}$ is a $p\times n$ matrix with components $d_{ij}$ . We have

\mathbf {D} =\mathbf {B} \mathbf {A} \implies d_{ij}=\sum _{k=1}^{m}b_{ik}a_{kj}

Clearly, $\mathbf {C} \neq \mathbf {D}$ in general, i.e., the matrix product is not commutative.

However, matrix multiplication is distributive. That means

\mathbf {A} (\mathbf {B} +\mathbf {C} )=\mathbf {A} \mathbf {B} +\mathbf {A} \mathbf {C} ~.

The product is also associative. That means

\mathbf {A} (\mathbf {B} \mathbf {C} )=(\mathbf {A} \mathbf {B} )\mathbf {C} ~.

Transpose of a matrix

Let $\mathbf {A}$ be a $m\times n$ matrix with components $a_{ij}$ . Then the transpose of the matrix is defined as the $n\times m$ matrix $\mathbf {B} =\mathbf {A} ^{T}$ with components $b_{ij}=a_{ji}$ . That is,

\mathbf {B} =\mathbf {A} ^{T}={\begin{bmatrix}a_{11}&a_{12}&a_{13}&\dots &a_{1n}\\a_{21}&a_{22}&a_{23}&\dots &a_{2n}\\a_{31}&a_{32}&a_{33}&\dots &a_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&a_{m3}&\dots &a_{mn}\end{bmatrix}}^{T}={\begin{bmatrix}a_{11}&a_{21}&a_{31}&\dots &a_{m1}\\a_{12}&a_{22}&a_{32}&\dots &a_{m2}\\a_{13}&a_{23}&a_{33}&\dots &a_{m3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\a_{1n}&a_{2n}&a_{3n}&\dots &a_{mn}\end{bmatrix}}

An important identity involving the transpose of matrices is

{(\mathbf {A} \mathbf {B} )^{T}=\mathbf {B} ^{T}\mathbf {A} ^{T}}~.

Inverse of a matrix

Let $\mathbf {A}$ be a $n\times n$ matrix. The inverse of $\mathbf {A}$ is denoted by $\mathbf {A} ^{-1}$ and is defined such that

{\mathbf {A} \mathbf {A} ^{-1}=\mathbf {I} }

where $\mathbf {I}$ is the $n\times n$ identity matrix.

The inverse exists only if $\det(\mathbf {A} )\neq 0$ . A singular matrix does not have an inverse.

An important identity involving the inverse is

{(\mathbf {A} \mathbf {B} )^{-1}=\mathbf {B} ^{-1}\mathbf {A} ^{-1},}

since this leads to: ${(\mathbf {A} \mathbf {B} )^{-1}(\mathbf {A} \mathbf {B} )=(\mathbf {B} ^{-1}\mathbf {A} ^{-1})(\mathbf {A} \mathbf {B} )=\mathbf {B} ^{-1}\mathbf {A} ^{-1}\mathbf {A} \mathbf {B} =\mathbf {B} ^{-1}(\mathbf {A} ^{-1}\mathbf {A} )\mathbf {B} =\mathbf {B} ^{-1}\mathbf {I} \mathbf {B} =\mathbf {B} ^{-1}\mathbf {B} =\mathbf {I} .}$

Some other identities involving the inverse of a matrix are given below.

The determinant of a matrix is equal to the multiplicative inverse of the

determinant of its inverse.

\det(\mathbf {A} )={\cfrac {1}{\det(\mathbf {A} ^{-1})}}~.

The determinant of a similarity transformation of a matrix

is equal to the original matrix.

\det(\mathbf {B} \mathbf {A} \mathbf {B} ^{-1})=\det(\mathbf {A} )~.

We usually use numerical methods such as Gaussian elimination to compute the inverse of a matrix.

Eigenvalues and eigenvectors

A thorough explanation of this material can be found at Eigenvalue, eigenvector and eigenspace. However, for further study, let us consider the following examples:

Let : $\mathbf {A} ={\begin{bmatrix}1&6\\5&2\end{bmatrix}},\mathbf {v} ={\begin{bmatrix}6\\-5\end{bmatrix}},\mathbf {t} ={\begin{bmatrix}7\\4\end{bmatrix}}~.$

Which vector is an eigenvector for $\mathbf {A}$ ?

We have $\mathbf {A} \mathbf {v} ={\begin{bmatrix}1&6\\5&2\end{bmatrix}}{\begin{bmatrix}6\\-5\end{bmatrix}}={\begin{bmatrix}-24\\20\end{bmatrix}}=-4{\begin{bmatrix}6\\-5\end{bmatrix}}$ , and $\mathbf {A} \mathbf {t} ={\begin{bmatrix}1&6\\5&2\end{bmatrix}}{\begin{bmatrix}7\\4\end{bmatrix}}={\begin{bmatrix}31\\43\end{bmatrix}}~.$

Thus, $\mathbf {v}$ is an eigenvector.

Is $\mathbf {u} ={\begin{bmatrix}1\\4\end{bmatrix}}$ an eigenvector for $\mathbf {A} ={\begin{bmatrix}-3&-3\\1&8\end{bmatrix}}$ ?

We have that since $\mathbf {A} \mathbf {u} ={\begin{bmatrix}-3&-3\\1&8\end{bmatrix}}{\begin{bmatrix}1\\4\end{bmatrix}}={\begin{bmatrix}-15\\33\end{bmatrix}}$ , $\mathbf {u} ={\begin{bmatrix}1\\4\end{bmatrix}}$ is not an eigenvector for $\mathbf {A} ={\begin{bmatrix}-3&-3\\1&8\end{bmatrix}}~.$

Reference

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