LaTeX - A Crash Course

Jai Ganesh · 2024-08-09 17:59:12

Matrix - X

Row operations

There are three types of row operations:
* row addition, that is adding a row to another.
* row multiplication, that is multiplying all entries of a row by a non-zero constant;
* row switching, that is interchanging two rows of a matrix;

These operations are used in several ways, including solving linear equations and finding matrix inverses.

Submatrix

A submatrix of a matrix is a matrix obtained by deleting any collection of rows and/or columns. For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2.

The minors and cofactors of a matrix are found by computing the determinant of certain submatrices.

A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain. Other authors define a principal submatrix as one in which the first k rows and columns, for some number k, are the ones that remain; this type of submatrix has also been called a leading principal submatrix.

Square matrix

A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries a ii form the main diagonal of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix.

Main types

Name : Example with n = 3

Diagonal matrix

{\displaystyle {\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\\\end{bmatrix}}}

gives

.

Lower triangular matrix

{\displaystyle {\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}}

gives

Upper triangular matrix

{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\\\end{bmatrix}}}

gives

.

Diagonal and triangular matrix

If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. Similarly, if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries outside the main diagonal are zero, A is called a diagonal matrix.

Identity matrix

The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example,

{\displaystyle \mathbf {I} _{1}={\begin{bmatrix}1\end{bmatrix}},\ \mathbf {I} _{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ \ldots ,\ \mathbf {I} _{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}}

gives

.

It is a square matrix of order n, and also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:

AI_n = I_mA = A

gives

for any m-by-n matrix A.

Jai Ganesh · 2024-08-11 18:12:57

Matrix - XI

Finding inverse matrix

{\displaystyle A\,\operatorname {adj} (A)=\operatorname {adj} (A)\,A=\det(A)I}

gives

where adj(A) denotes the adjugate matrix, det(A) is the determinant, and I is the identity matrix. If det(A) is nonzero, then the inverse matrix of A is

{\displaystyle A^{-1}={\frac {1}{\det(A)}}\operatorname {adj} (A).}

gives

.

This gives a formula for the inverse of A, provided

det(A) \ \neq \ 0

gives

.

In fact, this formula works whenever F is a commutative ring, provided that det(A) is a unit. If det(A) is not a unit, then A is not invertible over the ring (it may be invertible over a larger ring in which some non-unit elements of F may be invertible).

Jai Ganesh · 2024-08-13 00:05:52

Cramer's rule

Geometric interpretation - I

Cramer's rule has a geometric interpretation that can be considered also a proof or simply giving insight about its geometric nature. These geometric arguments work in general and not only in the case of two equations with two unknowns presented here.

Given the system of equations

{\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}&=b_{1}\\a_{21}x_{1}+a_{22}x_{2}&=b_{2}\end{matrix}}}

gives

it can be considered as an equation between vectors

{\displaystyle x_{1}{\binom {a_{11}}{a_{21}}}+x_{2}{\binom {a_{12}}{a_{22}}}={\binom {b_{1}}{b_{2}}}.}

gives

.

The area of the parallelogram determined by

{\displaystyle {\binom {a_{11}}{a_{21}}}}

gives

and

{\displaystyle {\binom {a_{12}}{a_{22}}}}

gives

is given by the determinant of the system of equations:

{\displaystyle {\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}.}

gives

.

Jai Ganesh · 2024-09-01 17:15:17

Cramer's rule

Geometric Interpretation - II

In general, when there are more variables and equations, the determinant of n vectors of length n will give the volume of the parallelepiped determined by those vectors in the n-th dimensional Euclidean space.

Therefore, the area of the parallelogram determined by

{\displaystyle x_{1}{\binom {a_{11}}{a_{21}}}}

gives

and

{\displaystyle {\binom {a_{12}}{a_{22}}}}

gives

has to be

{\displaystyle x_{1}}

gives

times the area of the first one since one of the sides has been multiplied by this factor. Now, this last parallelogram, by Cavalieri's principle, has the same area as the parallelogram determined by

{\displaystyle {\binom {b_{1}}{b_{2}}}=x_{1}{\binom {a_{11}}{a_{21}}}+x_{2}{\binom {a_{12}}{a_{22}}}}

gives

and

{\displaystyle {\binom {a_{12}}{a_{22}}}.}

gives

Equating the areas of this last and the second parallelogram gives the equation

{\displaystyle {\begin{vmatrix}b_{1}&a_{12}\\b_{2}&a_{22}\end{vmatrix}}={\begin{vmatrix}a_{11}x_{1}&a_{12}\\a_{21}x_{1}&a_{22}\end{vmatrix}}=x_{1}{\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}}

gives

.
from which Cramer's rule follows.

Math Is Fun Forum

#301 2024-08-09 17:59:12

Re: LaTeX - A Crash Course

#302 2024-08-11 18:12:57

Re: LaTeX - A Crash Course

#303 2024-08-13 00:05:52

Re: LaTeX - A Crash Course

#304 2024-09-01 17:15:17

Re: LaTeX - A Crash Course

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