# Notes on Mathematical Methods for Physicists Chapter2

本文最后更新于 2024年5月22日 晚上

# Notes on Mathematical Methods for Physicists

## Chapter2 Determinants & Matrices

Determinants

We begin the study of matrices by solving linear equations that will lead us to determinants and matrices. The concept of determinant and the notation were introduced by the renowned German mathematician and philosopher

Homogeneous Linear Equations

Suppose three unknowns

The problem is to determine under what conditions there is any solution , apart from the trivial one

If the volume spanned by

is not zero , then there is the only trivial solution

Conversely , if the aforementional determinant of the coefficient vanishes , then one of the row vectors is a combination of the other two.

This is **Cramer's Rule** for homogeneous linear equation.

Inhomogeneous Linear Equation

Simple example :

This is **Cramer's Rule** for inhomogeneous linear equation.

Definitions

Before defining a determinant , we need to introduce some related concepts and definitions.

**even** or **odd**. Thus a permutation can be identified as having either even or odd parity.

**Levi-Civita symbol** , which for an

We now define a determinant of order

The determinant

Properties of Determinants

Take determinants of order

Laplacian Development by Minor

The fact that a determinant of order

Linear Equation Systems

For equation

We define

Then we have

This is the **Cramer's Rule**.

If

Determinants & Linear Dependence

If the coefficients of

Linearly Dependent Equations

##### Situation

All the equations are homogeneous (which means all the right hand side quantities**manifold** (i.e. , a parameterized set) of solutions to our equation system.

##### Situation

A second case is where we have (or combine equations so that we have) the same linear form in two equations , but with different values of the right-hand quantities

##### Situation

A third , related case , is where we have a duplicated linear form , but with a common value of

Numerical Evaluation

There are many methods to evaluate determinants , even using computers. We use the **Gauss Elimination** to calculate determinants , which is a versatile procedure that can be used for evaluating determinants, for solving linear equation systems, and (as we will see later) even for matrix inversion.

Gauss Elimination : make the determinant into a form that all the entries in the lower triangle of the determinant. Then the only effective part is the product of thediagonal elements.

Matrices

Matrices are**matrix algebra**.

Basic Definitions

A matrix is a set of numbers or functions in a

A matrix for which**square**; One consisting of a single column (an**column vector** , while a matrix with only one row (therefore**row vector**.

Equality

If

Addition , Subtraction

Addition and subtraction are defined only for matrices**commutative** (**associative** (**null matrix** or **zero matrix** , can either be written as

Multiplication (by a Scalar)

Here we have

Note that the definition of multiplication by a scalar causes **each** element of marix

Matrix Multiplication (Inner Product)

**Matrix multiplication** is not an element-by-element operation like addition or multiplication by a scalar. The **inner product** of matrices

This definition causes the

It is useful to define the **commutator** of

which , as stated above , will in many cases be nonzero.

But , matrix multiplication is **associative** , meaning that

Unit Matrix

By direct matrix multiplication , it is possible to show that a square matrix with elements of value unity on its **principal diagonal** (the elements

note that it is not a matrix all of whose elements are unity. Giving such a matrix the name

Remember that

The previously introduced null matrices have only zero elements , so it is also obvious that for all

Diagonal Matrices

If a matrix**diagonal**.

Matrix Inverse

It will often be the case that given a square matrix**inverse** of

Every nonezero real (or complex) number

If**on the left** by

Since we started with a matrix**singular** , so our conclusion is that**on the right** by

This is inconsistent with the nonzero

The algebraic properties of real and complex numbers (including the existence of inverses for all nonzero numbers) define what mathematicians call a **field**. The properties we have identified for matrices are different ; they form what is called a **ring**.

A closed , but cumber-some formula for the inverse of a matrix exists ; it expresses the elements of

We describe here a well-known method that is computationally more efficient than the equation above , namely the Gauss-Jordan procedure.

##### Example Gauss-Jordan Matrix Inversion

The Gauss-Jordan method is based on the fact that there exist matrices

By using these transformations , the rows of a matrix can be altered (by matrix multiplication) in the same way as we did to the elements of determinants. If

What we need to do is to find out how to reduce

Write , side by side , the matrix

Multiply the rows as necessary to set to unity all elments of the first column of the left matrix ,

Subtracting the first row from the second an third rows , we obtain

Divide the second row by

Divide the third row by

Derivatives of Determinants

The formula giving the inverse of a matrix in terms of its minors enables us to write a compact formula for the derivative of a determinant

Applying now the chain rule to allow for the

Systems of Linear Equations

Note that if

This tells us two things : (a) that if we can evaluate

Then the result is important enough to be emphasized : **A square matrixis singular if and only if**.

Determinant Product Theorem

The Product Theorem is that

Note that

Rank of a Matrix

The concept of a matrix singularity can be refined by introducing the notion of the **rank** of a matrix. If the elements of a matrix are viewed as the coefficients of a set of linear forms , a square matrix is assigned a rank equal to the number of linearly independent forms that its elements describe. Thus , a nonsingular

Transpose , Adjoint , Trace

##### Transpose

The transpose of a matrix is the matrix that results from interchanging its row and column indices. This operation corresponds to subjecting the array to reflection about its principal diagonal. If a matrix is not square , its transpose will not even have the same shape as the original matrix. The transpose of

Note that transposition will convert a column vector into a row vector. A matrix that is unchanged by transposition is called **symmetric**.

##### Adjoint

The **adjoint** of a matrix

##### Trace

The **trace** , a quantity defined for square matrices , is the sum of the elements on the principal diagonal. Thus , for an

Some properties of the trace :

The second property holds even if

Considering the trace of the matrix product

Repeating this process , we also find

Operations on Matrix Products

There are some properties of the determinant and trace :

whether or not**not** a linear operator.

For other operations on matrix products , there are

Matrix Representation of Vectors

I have nothing to say , because it is easy to understand. (I am going to use

Orthogonal Matrices

A real matrix is termed **orthogonal** if its transpose is equal to its inverse. Thus , if

Since , for

Unitary Matrices

The definition is matrix which the adjoint is also the inverse is identified as **unitary**. One way of expressing this relationship is

If all the elements of a unitary matrix are real , the matrix is also orthogonal.

Since for any matrix

We observe that if

Hermitian Matrices

A matrix is identified as **Hermitian** , or , synonymously , **self-adjoint** , if it is equal to its adjoint. To be self-adjoint , a matrix

We see that the principal diagonal elements must be real.

Note that if two matrices**anti-Hermitian** , meaning that

Extraction of a Row or Column

It is useful to define column vectors