Use this calculator to find the diagonal matrix $D$, the invertible matrix $P$, and its inverse $P^{-1}$ for any given $2 \times 2$ matrix $A$, such that $A = P D P^{-1}$.
Diagonalize Matrix Calculator ($2 \times 2$)
Enter the elements of the $2 \times 2$ matrix $A$:
Calculation Successful: $A = P D P^{-1}$
The Diagonal Matrix $D$ and Invertible Matrix $P$ have been found.Diagonalize Matrix Calculator Formula
The goal of matrix diagonalization is to find matrices $P$ and $D$ such that:
A = P D P⁻¹
Where:
$A$: The original $n \times n$ matrix.
$D$: A diagonal matrix whose diagonal entries are the eigenvalues ($\lambda_i$) of $A$.
$P$: An invertible matrix whose columns are the corresponding eigenvectors ($v_i$) of $A$.
For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the process involves:
- Finding eigenvalues $\lambda$ by solving the characteristic equation: $$ \det(A – \lambda I) = 0 $$ $$ \lambda^2 – (a+d)\lambda + (ad-bc) = 0 $$
- For each eigenvalue $\lambda_i$, solving $(A – \lambda_i I)v_i = 0$ to find the corresponding eigenvector $v_i$.
Formula Sources: Wolfram MathWorld – Matrix Diagonalization, Paul’s Online Math Notes – Eigenvalues & Eigenvectors
Variables
- a11: The element in the first row, first column of matrix $A$.
- a12: The element in the first row, second column of matrix $A$.
- a21: The element in the second row, first column of matrix $A$.
- a22: The element in the second row, second column of matrix $A$.
Related Calculators
- Matrix Inverse Calculator
- Determinant Calculator ($3 \times 3$)
- Characteristic Polynomial Solver
- Vector Dot Product Calculator
What is Diagonalize Matrix Calculator?
Matrix diagonalization is a fundamental process in linear algebra that transforms a given square matrix $A$ into a similar diagonal matrix $D$. This transformation is incredibly useful because diagonal matrices are computationally simpler to work with, especially when calculating matrix powers (e.g., $A^n$), which are crucial in solving systems of differential equations, Markov chains, and analyzing stability in dynamical systems.
A matrix is diagonalizable if and only if there exists a full set of linearly independent eigenvectors. When diagonalization is possible, the diagonal entries of $D$ are the eigenvalues of $A$, and the columns of the transformation matrix $P$ are the corresponding eigenvectors. The relationship $A = P D P^{-1}$ effectively decomposes the matrix $A$ into components that describe scaling (eigenvalues in $D$) and change of basis (eigenvectors in $P$).
How to Calculate Diagonalize Matrix (Example)
Let’s use the example matrix: $A = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}$.
- Find the Characteristic Equation: $$ \det(A – \lambda I) = (3-\lambda)(2-\lambda) – (1)(0) = 0 $$ $$ \lambda^2 – 5\lambda + 6 = 0 $$
- Solve for Eigenvalues ($\lambda$): $$ (\lambda – 3)(\lambda – 2) = 0 $$ The eigenvalues are $\lambda_1 = 3$ and $\lambda_2 = 2$. These form the diagonal matrix $D$.
- Find Eigenvector for $\lambda_1 = 3$: $$ (A – 3I)v_1 = \begin{pmatrix} 0 & 1 \\ 0 & -1 \end{pmatrix} v_1 = 0 $$ This gives $0x + 1y = 0$, so $y=0$. A free variable $x=1$ gives the eigenvector $v_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$.
- Find Eigenvector for $\lambda_2 = 2$: $$ (A – 2I)v_2 = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} v_2 = 0 $$ This gives $1x + 1y = 0$, so $x = -y$. A free variable $y=1$ gives the eigenvector $v_2 = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$.
- Form $P$ and $D$: $$ D = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} \quad P = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} $$
- Verify: $A = P D P^{-1}$.
Frequently Asked Questions (FAQ)
Q: Can every matrix be diagonalized?
A: No. A matrix can only be diagonalized if it has $n$ linearly independent eigenvectors, where $n$ is the dimension of the matrix. Matrices with repeated eigenvalues might not be diagonalizable (e.g., $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$).
Q: What is the main application of diagonalization?
A: One of the most common applications is computing high powers of a matrix $A^k$. Since $A = P D P^{-1}$, we have $A^k = P D^k P^{-1}$. Calculating $D^k$ is trivial (just raise the diagonal entries to the power $k$), which simplifies the entire process immensely.
Q: What happens if the eigenvalues are complex?
A: If the matrix has complex eigenvalues, the resulting diagonal matrix $D$ and the transformation matrix $P$ will contain complex numbers. While mathematically valid, the matrix is not diagonalizable over the real numbers. This calculator only handles real-valued results.
Q: What is the relationship between diagonalization and similarity?
A: Two matrices $A$ and $B$ are similar if $B = P^{-1} A P$ for some invertible matrix $P$. Diagonalization is the process of finding a diagonal matrix $D$ that is similar to $A$, using the matrix $P$ formed by the eigenvectors.