S6 UNIT 7: Transformation of Matrices.

In linear algebra, a matrix transformation (often synonymous with linear transformation) is a special type of function that takes a vector as input and produces another vector as output, where this transformation can be represented by multiplication with a matrix. It’s a fundamental concept that bridges abstract vector spaces with concrete matrix operations.

What is a Matrix Transformation?

A transformation T: Rⁿ→ R^m is a matrix transformation if there exists an m×n matrix A such that for every vector x in Rⁿ,

                                                  T(x) = Ax

Here:

• x is a column vector in Rⁿ (n-dimensional input vector).
• A is an m×n matrix (the transformation matrix).
• Ax is the matrix-vector product, resulting in a column vector in R^m (m-dimensional output vector).

This means that any matrix multiplication can be interpreted as a geometric transformation of vectors.

Key Properties of Matrix Transformations (Linear Transformations)

Matrix transformations are a specific type of linear transformation. A transformation T is linear if it satisfies two properties:

1. 1. Additivity: T(u+v) =T(u)+T(v) for all vectors u, v in the domain.
   2. Homogeneity (Scalar Multiplication): T(cu)= cT(u) for all vectors u in the domain and all scalars c.

The kernel and range (also known as the null space and image, respectively) are two fundamental subspaces associated with a linear transformation. They provide crucial insights into how a transformation behaves, particularly concerning its injectivity (one-to-one) and surjectivity (onto) properties. 

Let $T:V\to W$ be a linear transformation from a vector space $V$ (the domain) to a vector space $W$ (the codomain).

The Kernel of a Transformation (Null Space).

The kernel of a linear transformation $T$, denoted as $\text{ker}\left(T\right)$ or $\text{Null}\left(T\right)$, is the set of all vectors in the domain $V$ that are mapped to the zero vector in the codomain $W$.

Formally: ker(T)= {v∈V ∣ T(v)=0_W​}

Where 0_W​ is the zero vector in the codomain W.

Key Properties of the Kernel:

• Subspace of the Domain: The kernel of a linear transformation is always a subspace of the domain V. This means it contains the zero vector of V, is closed under vector addition, and is closed under scalar multiplication.
• Injectivity (One-to-One): A linear transformation T is injective (one-to-one) if and only if its kernel contains only the zero vector, i.e., ker(T)={0_V​}. This means that distinct vectors in the domain always map to distinct vectors in the codomain.
• Relationship to Homogeneous Systems: If the linear transformation T is represented by a matrix A (i.e., T(x)=Ax), then finding the kernel of T is equivalent to solving the homogeneous system of linear equations Ax =0. The kernel is precisely the solution space of this system.
• Nullity: The dimension of the kernel of T is called the nullity of T, denoted as nullity(T) or dim(ker(T)).

How to Find the Kernel (for a matrix transformation T(x)= Ax)?

1. Set up the homogeneous equation Ax=0.
2. Form the augmented matrix [A∣0] and row-reduce it to its Reduced Row Echelon Form (RREF).
3. Write out the general solution to the system. The vectors that form the basis for this solution space constitute the basis for the kernel.

The Range of a Transformation (Image)

The range of a linear transformation T, denoted as range(T) or Im(T), is the set of all possible output vectors in the codomain W that can be obtained by applying T to vectors from the domain V.

Formally: range(T)= {w∈W ∣ ∃v∈V such that T(v)=w} This is equivalent to the set T(V)={T(v) ∣ v∈V}.

Key Properties of the Range:

• • • Subspace of the Codomain: The range of a linear transformation is always a subspace of the codomain W. It contains the zero vector of W, is closed under vector addition, and is closed under scalar multiplication.

  Surjectivity (Onto):

A linear transformation T is surjective (onto) if and only if its range is equal to the entire codomain W, i.e., range(T)=W. This means that every vector in the codomain can be reached by the transformation from some vector in the domain.

Relationship to Column Space: If the linear transformation T is represented by a matrix A (i.e., T(x)=Ax), then the range of T is precisely the column space of A (Col(A)). This is because any output Ax is a linear combination of the columns of A.Rank: The dimension of the range of T is called the rank of T, denoted as rank(T) or dim(range(T)). For a matrix A, rank(A)=dim(Col(A)).How to Find the Range (for a matrix transformation T(x)= Ax):

1. Identify the columns of the matrix A.
2. The range of T is the span of these column vectors.
3. To find a basis for the range, row-reduce the matrix A to its RREF. The pivot columns in the original matrix A (corresponding to the columns with leading ones in the RREF) form a basis for the column space, and thus for the range.

The Rank-Nullity Theorem

A fundamental theorem in linear algebra connects the dimensions of the kernel and range:

For a linear transformation T: V→W, where V is a finite-dimensional vector space:   

                                              dim(V)=dim(ker(T)) +dim(range(T))

Or, in terms of nullity and rank:

                                           dim(V)=nullity(T)+rank(T)

Diagonalization of a matrix is a powerful concept in linear algebra that simplifies complex matrix operations by transforming a square matrix into a special, much simpler form: a diagonal matrix. This transformation is achieved through a process involving eigenvalues and eigenvectors.

What is Diagonalization?

A square matrix A is said to be diagonalizable if it is similar to a diagonal matrix D. This means that there exists an invertible matrix P such that:

A=PDP⁻¹ or, equivalently: D=P⁻¹AP

Here:

• A is the original square matrix.
• D is a diagonal matrix, meaning all its entries are zero except for those on the main diagonal. The diagonal entries of D are the eigenvalues of A.
• P is an invertible matrix whose columns are the linearly independent eigenvectors of A, corresponding to the eigenvalues in D. This matrix P is often called the modal matrix or eigenvector matrix.
• P⁻¹ is the inverse of matrix P.

The expression P⁻¹AP is called a similarity transformation. If two matrices are similar, they share many important properties, including their eigenvalues, determinant, trace, and rank.

Conditions for Diagonalizability

Not all square matrices are diagonalizable. A matrix A is diagonalizable if and only if:

1. It has a full set of linearly independent eigenvectors. For an n×n matrix, this means it must have n linearly independent eigenvectors.
2. Equivalently, for each eigenvalue, its geometric multiplicity (the dimension of its eigenspace) must be equal to its algebraic multiplicity (the number of times it appears as a root of the characteristic polynomial).

Important Note:

• If a matrix has n distinct eigenvalues, it is guaranteed to be diagonalizable.
• If a matrix has repeated eigenvalues, it may or may not be diagonalizable. You need to check if you can find enough linearly independent eigenvectors for each repeated eigenvalue. If not, the matrix is defective and not diagonalizable. In such cases, a matrix can sometimes be put into a Jordan Canonical Form, which is the “next best thing” to a diagonal form.

How to Diagonalize a Matrix (Recipe)

To diagonalize a matrix A:

1. Find the Eigenvalues of A: Solve the characteristic equation det(A−λI) =0 for λ. These λ values are the eigenvalues.
2. Find the Eigenvectors for Each Eigenvalue: For each eigenvalue λ, solve the system (A−λI)x =0 to find the corresponding eigenvectors x.
3. Form the Matrix P: If you found n linearly independent eigenvectors, arrange them as the columns of a matrix P.

4. Form the Diagonal Matrix D: Create a diagonal matrix D with the eigenvalues on its main diagonal, in the same order as their corresponding eigenvectors appear in P.
5. Verify (Optional but Recommended): Calculate P⁻¹AP to confirm that it equals D

The Process of Diagonalization:

• The Diagonalization Formula: Students will learn the fundamental relationship A=PDP⁻¹ (or D=P⁻¹AP) and its components:
  • A: The original square matrix to be diagonalized.
  • D: A diagonal matrix whose diagonal entries are the eigenvalues of A.
  • P: An invertible matrix whose columns are the linearly independent eigenvectors of A, ordered to correspond with the eigenvalues in D.
• Steps to Diagonalize a Matrix: A clear, step-by-step procedure:

1. Calculate the eigenvalues of A.
2. For each eigenvalue, find a basis for its corresponding eigenspace (i.e., find the eigenvectors).
3. Check for diagonalizability: If there are n linearly independent eigenvectors for an n×n matrix, it is diagonalizable. Otherwise, it is not.
4. Construct P using the eigenvectors as columns.
5. Construct D using the eigenvalues on the diagonal (in the same order as their eigenvectors in P).
6. (Optional but good for understanding) Verify the relationship A=PDP⁻¹ or D=P⁻¹AP.

Conditions for Diagonalizability:

• Sufficient Condition: If an n×n matrix has n distinct eigenvalues, it is guaranteed to be diagonalizable.
• Necessary and Sufficient Condition: A matrix is diagonalizable if and only if for every eigenvalue, its algebraic multiplicity equals its geometric multiplicity.
• Non-Diagonalizable (Defective) Matrices: Students will learn that some matrices cannot be diagonalized (they are “defective”) because they don’t have enough linearly independent eigenvectors. They might be introduced to the concept of the Jordan Canonical Form as a “next best” decomposition for such matrices.