Diagonalize Matrix Calculator

The Diagonalize Matrix Calculator transforms a square matrix into its diagonal form through eigenvalue and eigenvector computations and simplifies complex linear algebra tasks.

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Introduction to Diagonalize Matrix Calculator:

Diagonalize Matrix Calculator is an online tool that allows you to evaluate the square matrix if it is diagonalizable. Our tool is efficient enough to check whether the matrix is diagonalizable or not if it is then it provides the diagonal form of the matrix along with the corresponding transform matrix.

Diagonalize Matrix Calculator with steps

This online matrix diagonalize calculator is valuable for students, researchers, and professionals because it is used in various fields where matrix analysis is required.

What is a Diagonal Matrix?

The diagonal matrix is the matrix that has all zero entries except the main diagonal values from the top left slide to the bottom right side inside the matrix.

Diagonalization of a matrix is a process in which a square matrix is changed into a diagonal matrix after finding the value of eigenvectors in linear algebra. It simplifies calculations involving matrices, especially when dealing with repeated operations or powers of the matrix.

How to Calculate the Diagonal Matrix?

To calculate the diagonal matrix of a square matrix A, first, the diagonalization matrix calculator checks whether the given matrix is linearly independent or not. If it is linearly independent then you can say it is diagonalized otherwise it is not diagonalized. The steps are:

To find the Eigenvalues and Eigenvectors, the matrix diagonalizable calculator computes the eigenvalues (λ) of the matrix A and converts it into the characteristic equation det(A - λI) = 0, where I is the identity matrix.

Step 1:

For each eigenvalue, solve the equation (A - λI)v = 0 to find the corresponding eigenvectors (v).

Step 2:

Then use the eigenvalues to find the eigenvector basis for matrix P.

Step 3:

Arrange the eigenvector values that are present in a column into a matrix P. Make sure the eigenvectors correspond to the eigenvalues in the same order.

Step 4:

Compute PDP^(-1). If A = PDP^(-1), to check whether they get a diagonal matrix or not if you get a diagonal matrix then the matrix A is diagonalized.

Let's see this process with an example of a 2 by 2 and 3 by 3 matrix to get conceptual clarity.

How to Diagonalize a 2x2 Matrix?

In order to find the diagonal of the 2 by 2 matrix, the diagonalize matrix calculator finds PAP^-1.

$$ A \;=\; \biggr[\begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \biggr] $$

Solution:

Find the eigenvalues using this equation det(𝜆𝐼−𝐴)=0 and we get the value of 𝜆 such as:

$$ \lambda_1 \;=\; 1, \lambda_2 \;=\; 1 $$

Now again add the λ value in the above equation one by one for λ=1.

To find the eigenvectors, solve the equation(λI - A)X = 0.

$$ \biggr[\begin{matrix}\lambda - 1 & -1 \\ 0 & \lambda - 1 \\ \end{matrix} \biggr] $$

$$ \biggr[\begin{matrix} 1 - 1 & -1 \\ 0 & 1 - 1 \\ \end{matrix} \biggr] \;=\; \biggr[\begin{matrix} 0 & -1 \\ 0 & 0 \\ \end{matrix} \biggr] $$

Then the eigenvector is

$$ t \biggr[\begin{matrix} 1 \\ 0 \\ \end{matrix} \biggr] $$

Eigenvector eigen basis is

$$ X_1 \;=\; \biggr[\begin{matrix} 1 \\ 0 \\ \end{matrix} \biggr] $$

In this case, we have one eigenvector basis for two multiplicities. we need a 2 by-2 matrix to construct a P matrix. Due to one value of the eigenvector basis, we cannot find the diagonal of a given matrix.

How to Diagonalize a 3x3 Matrix?

In order to find the diagonal of the 3 by 3 matrix, the diagonalize the matrix calculator determines the diagonal of the matrix,

$$ A \;=\; \biggr[\begin{matrix} 2 & 0 & 0 \\ 1 & 4 & -1 \\ -2 & -4 & 4 \\ \end{matrix} \biggr] $$

Solution:

For the diagonal of the matrix,

$$ P^{-1} AP \;=\; D $$

Find the eigenvalues using this equation det(𝜆𝐼−𝐴)=0

$$ det \left( \lambda \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] - \left[ \begin{matrix} 2 & 0 & 0 \\ 1 & 4 & -1 \\ -2 & -4 & 4 \\ \end{matrix} \right] \right) \;=\; 0 $$

After solving the above condition, the values of λ is

$$ \lambda_1 \;=\; 2,\; \lambda_2 \;=\; 2,\;and\; \lambda_3 \;=\; 6 $$

Now again add the λ value in the above equation one by one for λ=2

Find the eigenvectors for solving (2I - A)X = 0,

$$ t \left[ \begin{matrix} -2 \\ 1 \\ 0 \\ \end{matrix} \right] + s \left[ \begin{matrix} 1 \\ 0 \\ 1 \\ \end{matrix} \right] $$

After simplification, we get the values of X1 and X2,

$$ X_1 \;=\; \left[ \begin{matrix} -2 \\ 1 \\ 0 \\ \end{matrix} \right] + X_2 \;=\; \left[ \begin{matrix} 1 \\ 0 \\ 1 \\ \end{matrix} \right] $$

For λ=6 the value of X3 becomes,

$$ X_3 \;=\; \left[ \begin{matrix} 0 \\ 1 \\ -2 \\ \end{matrix} \right] $$

Combine the values of X1, X2, and X3 to make a matrix that is called P.

$$ P \;=\; [\begin{matrix} X_1 & X_2 & X_3 \\ \end{matrix} ] \;=\; \left[ \begin{matrix} -2 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & -2 \\ \end{matrix} \right] $$

For P^-1 value, after solving the P^-1 value with the help of this formula we get P^-1= Adjoint P/det (P).

$$ P^{-1} \;=\; \left[ \begin{matrix} -¼ & ½ & ¼ \\ ½ & 1 & ½ \\ ¼ & ½ & -¼ \\ \end{matrix} \right] $$

Combine P, P^-1, and A matrix to make them a diagonal matrix such as

$$ P^{AP} \;=\; \left[ \begin{matrix} -¼ & ½ & ¼ \\ ½ & 1 & ½ \\ ¼ & ½ & -¼ \\ \end{matrix} \right] \left[ \begin{matrix} 2 & 0 & 0 \\ 1 & 4 & -1 \\ -2 & -4 & 4 \\ \end{matrix} \right] \left[ \begin{matrix} -2 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & -2 \\ \end{matrix} \right] $$

$$ =\; \left[ \begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 6 \\ \end{matrix} \right] $$

How to Use Diagonalize Matrix Calculator?

The matrix diagonalize calculator has a simple design that helps everyone to solve the given matrix diagonalize question instantly. You just need to put your problem in it and follow some important instructions to get results of the diagonalized matrix without any hustle. These instructions are:

Choose the size of the matrix that is given in the input box of the matrix diagonalization calculator.
Enter the elements of the matrix in the input fields that you want to diagonalize in the solution.
Review your given input value to get the exact solution of the diagonalized matrix question.
The Calculate button evaluates the example of matrix diagonalization problems.
If you want to check the diagonalization matrix calculator then you can use the load example and get its solution to know more about this concept.
Click the “Recalculate” button for the evaluation of more examples of the diagonalized matrix with the solution.

Outcome from Matrix Diagonalize Calculator:

Diagonalize Matrix Calculator provides you with a solution as per your input problem when you click on the calculate button. It may include as:

In the Result Box,

Click on the result button so that the Diagonalize Matrix question you get the solution.

Steps Box
When you click on the steps option of the matrix diagonalization calculator, you get the solution of Diagonalize Matrix questions in a step-by-step process.

Benefits of Diagonalization Matrix Calculator:

The diagonal matrix calculator has many benefits that you obtain when using it to solve Diagonalize Matrix problems to get its solution. Our tool only gets the input value and provides a solution. These benefits are

The diagonalize the matrix calculator is a reliable tool as it always provides you with accurate solutions of the given matrix into a diagonalized matrix.
It is an efficient tool that provides solutions in the form of diagonalized matrix from the given matrix problems in a few seconds.
The matrix diagonalizable calculator is a learning tool that helps parents to teach their children about the concept of diagonalized matrix very easily through online platforms.
It is a handy tool that solves diagonalized matrix problems quickly and you do not put any type of external effort.
Matrix diagonalize calculator is a free tool that allows you to use it for the calculation of diagonalized matrix without taking a fee.
Diagonalize Matrix Calculator is an easy-to-use tool, anyone or even a beginner can easily use it for the solution of matrix diagonalize problems.

Related References

What is diagonalize matrix? Explain
Give some examples of diagonalization of matrix:
How to evaluate matrix diagonalize?
An overview of matrix diagonalization:

Frequently Ask Questions

Can the upper triangular matrix have zero on the diagonal?

Yes, an upper triangular matrix can have zeros on the diagonal. Upper triangular matrices are defined as all elements below the main diagonal (including some part of the diagonal itself) equal to zero.

In this matrix, all elements below the main diagonal are zero, including the diagonal elements then the matrix is still considered an upper triangular because of its definition. All non-zero elements are located above or on the main diagonal.

Is the zero matrix diagonal?

Yes, the zero matrix has a diagonal. A diagonal matrix is a square matrix where all the elements of the main diagonal are zero. Since the zero matrix has all its elements set to zero, including those on the main diagonal, it meets the criteria of a diagonal matrix.

For example, the 3x3 zero matrix is:

$$ \left( \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right) $$

This matrix has zeros everywhere, including the diagonal, so it is indeed a diagonal matrix.

Does a diagonal matrix have to be square?

Diagonal matrix is not necessarily a square only but a nonsquare matrix also have diagonal. A diagonal matrix can be rectangular if it follows the criteria of having non-zero elements only on the main diagonal, and all other elements outside the main diagonal are zero.

In a square matrix, where the number of rows is equal to the number of columns, the diagonal matrix is often represented as:

$$ \left( \begin{matrix} d_1 & 0 & 0 & \cdots & 0 \\ 0 & d_2 & 0 & \cdots & 0 \\ 0 & 0 & d_3 & \cdots & 0 \\ 0 & 0 & 0 & \cdots & d_n \\ \end{matrix} \right) $$

Here, d1, d2,…, dn are the diagonal elements, and the matrix is square. However, since the diagonal matrix is not always square, it may have fewer rows than columns or fewer columns than rows.

Why do we diagonalize a matrix?

Diagonalizing a matrix has some important purposes in mathematics that are applied in various fields that show why diagonalization is useful:

Diagonal matrices simplify the row operation and make computation easier such as in matrix multiplication, exponentiation, and solving linear systems.
It helps to understand the Matrix properties specially in fields of quantum mechanics, control theory, and signal processing.
Diagonalization can simplify the solution of systems of linear differential equations.
Diagonalization is a necessary step in spectral decomposition, where a matrix is expressed as a sum of matrices, as a mode of operation in the applications of signal processing.

When can a matrix not be diagonalized?

A matrix cannot be diagonalized if certain criteria are not fulfilled. The most common reason a matrix cannot be diagonalized is that the set of eigenvectors is not linearly independent. It has many reasons why a matrix cannot be diagonalized:

when the matrix has repeated eigenvalues but not enough linearly independent eigenvectors that are associated with each eigenvalue.

The matrices may have repeated eigenvalues and only a few are linearly independent eigenvectors than expected ones based on their algebraic multiplicity.

Even though a matrix cannot be diagonalized, other methods such as Jordan canonical form or singular value decomposition may be used for the solution.

Amanda Jepson

Last Updated February 21, 2024