Linear Algebra Specialist & Data Analyst
Effortlessly perform matrix decomposition with our professional matrix diagonalisation calculator. This tool finds eigenvalues and eigenvectors to construct the diagonal matrix $D$ and invertible matrix $P$.
Matrix Diagonalisation Calculator
matrix diagonalisation calculator Formula
Where A is the original matrix, P is the matrix of eigenvectors, and D is the diagonal matrix of eigenvalues.
Source: Wolfram MathWorld – Diagonalization | Wikipedia – Diagonalizable Matrix
Variables:
- a11, a12, a21, a22: The elements of the input square matrix $A$.
- Eigenvalues (λ): Scalars that satisfy the equation $\det(A – \lambda I) = 0$.
- Eigenvectors (v): Non-zero vectors that satisfy $Av = \lambda v$.
- Matrix P: The modal matrix composed of eigenvectors as columns.
- Matrix D: The diagonal matrix containing eigenvalues on the main diagonal.
Related Calculators:
- Matrix Determinant Calculator
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- Matrix Inverse Calculator
- Characteristic Polynomial Calculator
What is Matrix Diagonalisation?
Matrix diagonalization is the process of finding a diagonal matrix that is similar to a given square matrix. This process is fundamental in linear algebra because it simplifies complex operations like matrix exponentiation and solving systems of linear differential equations.
A matrix is diagonalizable if and only if it has enough linearly independent eigenvectors to form a basis. For an $n \times n$ matrix, this means having $n$ linearly independent eigenvectors. Not all matrices are diagonalizable; some “defective” matrices cannot be reduced to diagonal form.
How to Calculate Matrix Diagonalisation (Example)
Consider a 2×2 matrix A: [4, 1; 2, 3]
- Find the Characteristic Equation: $\det(A – \lambda I) = (4-\lambda)(3-\lambda) – (1)(2) = 0$.
- Solve for $\lambda$: $\lambda^2 – 7\lambda + 10 = 0$, giving eigenvalues $\lambda_1 = 5, \lambda_2 = 2$.
- Find Eigenvectors: For $\lambda = 5$, solve $(A-5I)v=0$ to get $v_1 = [1, 1]$. For $\lambda = 2$, get $v_2 = [1, -2]$.
- Form P and D: $P = [[1, 1], [1, -2]]$ and $D = [[5, 0], [0, 2]]$.
- Verify: $A = PDP^{-1}$.
Frequently Asked Questions (FAQ)
What if the eigenvalues are complex?
If the characteristic equation has no real roots, the matrix is not diagonalizable over the real numbers, though it may be over complex numbers.
Is every square matrix diagonalizable?
No. A square matrix is only diagonalizable if the geometric multiplicity of each eigenvalue equals its algebraic multiplicity.
Why is diagonalization useful?
It allows us to compute $A^n$ easily by calculating $P D^n P^{-1}$, which is much faster than repeated matrix multiplication.
What is a diagonal matrix?
A diagonal matrix is a square matrix where all entries outside the main diagonal are zero.