## Linear Maps Between Vector Spaces: Additional Definitions and Properties of Linear Maps

#### Learning Outcomes

• Compute the transpose of a matrix. Compute the transpose of the product of two matrices
• Describe the relationship between invertibility, the existence of an inverse, and the determinant of a matrix. Compute the inverse of a 2×2 and a 3×3 matrix
• Describe of invariants as quantities that are the same under an orthonormal change of basis. Compute the first invariant, the second invariant, and the third invariant of a 3×3 matrix

### Matrix Transpose

Let be a linear map. If is the matrix representation after choosing particular orthonormal basis sets for the underlying spaces, then, the transpose of or , is a map whose columns are the rows of . In component form, this means:

The above definition relies on components. Another equivalent but more convenient definition is as follows.
Let be a linear map. Then, is the unique linear map that satisfies:

Any of the above two definitions can be used to show the following facts about the transpose of square matrices.

Notice that since the determinant of a square matrix is the same whether we consider the rows or the columns, then:

For example

### Matrix Inverse

Let be a linear map. The following are all equivalent:

• is invertible
• The rows of the matrix representation of are linearly independent
• The kernel of the contains only the zero vector
In this case, the inverse of is denoted and satisfies:

Notice that is unique, because if there is another matrix such that , then .
Notice also that if such that and , then, .

If the linear maps and are invertible, then it is easy to show that is also invertible and:

#### Matrix Inverse in

Consider the matrix:

Then, the inverse of can be shown to be:

Try it out, input the values of the matrix and press evaluate to calculate its inverse.

#### Matrix Inverse in

Consider the matrix:

If and , then, the inverse of can be shown to be:

Try it out, input the values of the matrix and press evaluate to calculate its inverse.

### Invariants

Consider with the two orthonormal basis sets and with a coordinate transformation matrix such that .
Clearly, the components of vectors and the matrices representing linear operators change according to the chosen coordinate system (basis set). Invariants are functions of these components that do not change whether or is chosen as the basis set.
The invariants usually rely on the fact that .

#### Vector Invariants

##### Vector Norm

A vector has the representation with components when is the basis set. Alternatively, it has the representation with components when is the basis set.
The norm of the vector u is an invariant since it is equal whether we use or .
The norm of when is the basis set:

The norm of is also equal to the norm of :

##### Vector Dot Product

Similarly, the dot product between two vectors is invariant:

#### Matrix Invariants in

We will restrict our discussion of invariants when the underlying space is . A linear operator has the matrix representation with components when is the basis set. Alternatively, it has the representation with components when is the basis set. The following are some invariants of the matrix :

##### First Invariant, Trace

The trace of or is defined as:

is invariant for if we consider the components in :

It is straight forward from the definition to show that ,:

The above definition for the first invariant depends on the components in a given coordinate system. Another definition according to P. Chadwick that is independent of a coordinate system is given as follows:

where, and are three arbitrary linearly independent vectors. Use the components of and to verify that the two definitions are equivalent.

##### Second Invariant

The second invariant is defined as:

Clearly, since is invariant, so is :

Another definition for the second invariant according to P. Chadwick that is independent of a coordinate system is given as follows:

where, and are three arbitrary linearly independent vectors. Use the components of and to verify that the two definitions are equivalent.

##### Third Invariant, the Determinant

The third invariant is defined as the determinant of the matrix ;

Clearly, is invariant:

Another definition for the third invariant according to P. Chadwick that is independent of a coordinate system is given as follows:

where, and are three arbitrary linearly independent vectors. Use the components of and to verify that the two definitions are equivalent.
The trace (first invariant) and determinant (third invariant) of a matrix are related as follows:

##### Eigenvalues Are Invariants

The eigenvalues of the matrices and are the same (why?).
It is worth mentioning that the three invariants mentioned above appear naturally in the characteristic equation of :

Input the components of a matrix in the following tool and three angles for coordinate transformation. The tool then calculates the three matrix invariants along with the eigenvalues and eigenvectors in both coordinate systems. As expected, the invariants and the eigenvalues are the same. However, the components of the eigenvectors are different. The vectors themselves are the same, but the components are different according to the relationship .

### Cayley-Hamilton Theorem

The Cayley-Hamilton Theorem is an important theorem in linear algebra that asserts that a matrix satisfies its characteristic equation. In other words, let . The eigenvalues of are those that satisfy:

where are polynomial expressions of the entries of the matrix . In particular, . Then, the Cayley-Hamilton Theorem asserts that:

The first equation is a scalar equation which is a polynomial expression of the variable . However, the second equation is a matrix equation in which the sum of the given matrices gives the 0 matrix. Without attempting a formal proof for the theorem, in the following we will show how the theorem applies to and .

#### Two Dimensional Matrices

Consider the matrix:

Therefore, the characteristic equation of is given by:

I.e.,

The matrix satisfies the characteristic equation as follows:

Where:

and

The following Mathematica code illustrates the above expressions.

View Mathematica Code

M = {{M11, M12}, {M21, M22}}
A = M.M – Tr[M] M + Det[M]*IdentityMatrix[2] FullSimplify[A]

#### Three Dimensional Matrices

Consider the matrix:

Therefore, the characteristic equation of is given by:

I.e.,

The matrix satisfies the characteristic equation as follows:

The above polynomial expressions in the components of the matrix equate to zero as illustrated using the following Mathematica code:

View Mathematica Code

M = {{M11, M12,M13}, {M21, M22,M23},{M31, M32,M33}}
I2=1/2*(Tr[M]^2-Tr[M.M]);
A = M.M.M – Tr[M] M.M + I2*M-Det[M]*IdentityMatrix[3] FullSimplify[A]

One can show using induction that for , the matrix for can be written as a linear combination of , and such that:

where , and are functions of the invariants , and .
Similarly, if is invertible, then, for can be written as:

where , and are functions of the invariants , and .

To calculate the eigenvalues and eigenvectors of a real-values 3-by-3 matrix, the cubic characteristic equation of the matrix should be solved for its roots. The roots of characteristic equation

can be determined through factoring the terms of the polynomial, or guessing the roots. However, factoring the polynomial or/and guessing the roots may not be easy and straight forward for all cases. The following general method can be recruited to find the real roots of the characteristic (or any) cubic polynomial.

For the sake of simplicity of the formulations, we write the characteristic equation as,

where and are real values.

This cubic equation can have both real and complex roots. We consider only the real roots leading to real eigenvalues and eigenvectors of .

Finding the roots is commenced by calculating the discriminant of the polynomial as,

Once the discriminant is calculated, the following statements hold,

1- If , the polynomial has three distinct real roots. The roots can be calculated by direct factoring the polynomial (if easy enough) or determined as follows.

where .

2- If , the polynomial has one real root and two complex roots. The real root can be computed by factoring the polynomial or,

where,

3- If and then the polynomial has three repeated roots, i.e. a triple root, as

The polynomial with three repeated roots can be factored as .

4- If and then the polynomial has two repeated roots, i.e. a double root, as,

and a simple (not repeated) root,

In this case, the polynomial can be factored as .

Once the eigenvalues (the roots) are calculated, the eigenvectors can be calculated by solving the indeterminate and homogeneous linear system for . This linear system can have the following three types of general sets of solutions as the eigenvectors of .

1- All eigenvectors associated with an eigenvalue can be written as the scalar product of any eigenvector of , i.e. the eigenvectors belong to a one-dimensional subspace of . In this case, if is an eigenvalue of and is its associated eigenvector, then describes all the eigenvectors associated with \. For example,

has three eigenvalues and eigenvectors as the following general solutions (associated with the eigenvalues),

where . As we can see, each eigenvector belongs to a one-dimensional subspace of . Instances of the eigenvectors for are,

2- All eigenvector associated with an eigenvalue belong to a two-dimensional subspace of . It means if is an eigenvalue of , there are two linearly independent eigenvectors and such that , i.e. is a linear combination of and . Note that and are also associated with . For example,

has an eigenvalue and its associated eigenvectors are all vectors expressed as,

The above is the general solution of . An instance of eigenvector of can be given by setting values for and . For example,

for and .

3- For an eigenvalue , all vectors in are eigenvectors of . In other words, is the set of general solution of and eigenvectors of . For example, the identity matrix

has a single eigenvalue and all vectors in are its eigenvectors.

The following interactive tool calculates real eigenvalues and eigenvectors of a three dimensional matrix. It shows the types of the roots and general solutions regarding the eigenvectors. Note: Simple, double, and triple roots are referred to as non-repeated, twice repeated, and three-time repeated roots respectively. The eigenvalues and the eigenvectors are listed in the same order. The precision of outputs is .