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Linear Algebra: Help & Tutorials6 chapters | 44 lessons

Instructor:
*Paul Bohan-Broderick*

Paul has been teaching many subjects in many different ways since he received his PhD in 2001.

Diagonalization is the process of transforming a matrix into diagonal form. Diagonal matrices represent the eigenvalues of a matrix in a clear manner. This lesson will focus on finding the diagonalized form of a simple matrix.

A diagonal matrix is a matrix in which non-zero values appear only on its main diagonal. In other words, every entry not on the diagonal is 0. Diagonalization is the process of transforming a matrix into diagonal form.

Not all matrices can be diagonalized. A diagonalizable matrix could be transformed into a diagonal form through a series of basic operations (multiplication, division, transposition and so on). However, this process can be long and is not easily described. Fortunately, diagonalization can be carried out through a more general algorithm that takes advantage of the matrix's **characteristic polynomial**.

Since eigenvectors and eigenvalues of a matrix are so important for understanding the how and why of a diagonal matrix, it would be worthwhile to quickly review them here. A matrix, say *A*, can be understood to represent a function or transformation that could be applied to a vector, say *v*. If the vector changes in magnitude, but does not otherwise transform, then it is an eigenvector of the matrix. Whatever value the vector is changed by is called the eigenvalue of the matrix. Usually the scalar number by which the vector is multiplied is called lambda. This relationship can be expressed through the following equation:

The diagonalized matrix is not in the same vector space as the original matrix. The eigenvectors of the matrix will be the basis of the new space. When a matrix has been diagonalized, the columns of each corresponds to an eigenvector of the matrix and each value (one per column) represent the eigenvalues of the matrix.

Matrix diagonalization is usually carried out by manipulating the characteristic polynomial of the matrix. The eigenvalues of the matrix are the zeros of the characteristic polynomial.

Not every matrix can be diagonalized.

Every square matrix has a characteristic equation. However, the values of the diagonalized matrix are the values of lambda when P(lambda) = 0. If the determinant of the characteristic function is not 0 for some lambda, then the matrix cannot be diagonalized.

The determinant of the square matrix is a useful property. The algorithm for computing the determinant gets more complicated as the matrix gets larger. For instance, the determinant of the 2x2 matrix

Larger matrices require longer and more complicated computations.

The inner expression of the determinant equation is a new matrix created by multiplying the identity matrix by lambda and subtracting the original matrix by lambda. Taking the determinant of this new matrix will yield a polynomial. The zeros of that polynomial are the entries of the diagonal matrix. In effect, this new matrix is created by subtracting lambda from each entry on the diagonal of the matrix.

The characteristic polynomial of a matrix will generally be the same order as the size of the matrix. In other words, a 2x2 matrix will have a second order (binomial) characteristic equation. A 3x3 matrix will have a third order equation. The largest term will include the cube of lambda and so on.

It is worth repeating: The new, diagonal matrix created by this procedure is really a representation of the original matrix. However, it is now represented in a new vector space. The eigenvectors of the matrix are the basis of this new space.

Computing characteristic functions is a very involved computation for any matrix above 3x3. So, our example will focus on a 2x2 matrix.

Subtract lambda from the diagonal:

Take the determinant:

Clean this up to derive the characteristic polynomial:

Solving for lambda, the eigenvalues of the original matrix are 2 and 3. This yields a new diagonal matrix:

A **diagonal matrix** is a special form of a square matrix in which non-zero entries only appear on the major diagonal of the matrix. Diagonal matrices are useful because the are easier to perform calculations on, and more importantly, because they present an explicit connection between the **eigenvectors** and **eigenvalues** of the original matrix. The process of diagonalization involves constructing and solving the **charactersitic polynomial** of the matrix. The solutions of this polynomial are the entries of the diagonal matrix.

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Linear Algebra: Help & Tutorials6 chapters | 44 lessons

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