Find Matrix From Eigenvalues And Eigenvectors Calculator

Matrix Reconstruction Calculator

Enter the eigenvalues and corresponding eigenvectors for a 2×2 matrix to reconstruct the original matrix A.

What is Finding a Matrix from Eigenvalues and Eigenvectors?

Finding a matrix from its eigenvalues and eigenvectors is the process of reconstructing a square matrix (in this calculator, a 2×2 matrix) when you know its eigenvalues and the corresponding eigenvectors. This process is essentially the reverse of finding eigenvalues and eigenvectors for a given matrix. It relies on the concept of matrix diagonalization.

If a matrix A has a full set of linearly independent eigenvectors, it can be diagonalized. This means it can be expressed as A = PDP^-1, where P is the matrix whose columns are the eigenvectors of A, and D is the diagonal matrix whose diagonal entries are the corresponding eigenvalues of A.

Who should use it?

This calculator and process are useful for students learning linear algebra, engineers, physicists, data scientists, and anyone working with matrix transformations, systems of differential equations, or principal component analysis (PCA), where understanding the relationship between a matrix, its eigenvalues, and eigenvectors is crucial.

Common Misconceptions

A common misconception is that any matrix can be reconstructed this way. This is only true if the matrix is diagonalizable, which means it must have a full set of linearly independent eigenvectors. For an n x n matrix, you need n linearly independent eigenvectors. If the eigenvectors are linearly dependent, the matrix P is singular (its determinant is zero), and P^-1 does not exist, so A cannot be uniquely determined from A = PDP^-1 in this manner.

Find Matrix from Eigenvalues and Eigenvectors Calculator: Formula and Mathematical Explanation

Given a 2×2 matrix A with two eigenvalues λ₁, λ₂ and corresponding linearly independent eigenvectors v₁ = [x₁, y₁]^T, v₂ = [x₂, y₂]^T, we can reconstruct A.

1. Form the matrix P whose columns are the eigenvectors: P = [v₁ v₂] = [[x₁, x₂], [y₁, y₂]]

2. Form the diagonal matrix D with eigenvalues on the diagonal: D = [[λ₁, 0], [0, λ₂]]

3. Calculate the inverse of P, P^-1. For a 2×2 matrix P = [[a, b], [c, d]], P^-1 = (1/(ad-bc)) * [[d, -b], [-c, a]]. Here, det(P) = x₁y₂ – x₂y₁. If det(P) ≠ 0, then: P^-1 = (1/det(P)) * [[y₂, -x₂], [-y₁, x₁]]

4. The original matrix A is given by A = PDP^-1. First, calculate PD: PD = [[λ₁x₁, λ₂x₂], [λ₁y₁, λ₂y₂]]

Then, A = (PD)P^-1: A = (1/det(P)) * [[(λ₁x₁y₂ – λ₂x₂y₁), (λ₂x₁x₂ – λ₁x₁x₂)], [(λ₁y₁y₂ – λ₂y₂y₁), (λ₂x₁y₂ – λ₁y₁x₂? No… -λ₁y₁x₂ + λ₂y₂x₁)]]

A₁₁ = (λ₁x₁y₂ – λ₂x₂y₁) / det(P)
A₁₂ = (λ₂x₁x₂ – λ₁x₁x₂) / det(P)
A₂₁ = (λ₁y₁y₂ – λ₂y₂y₁) / det(P)
A₂₂ = (λ₂x₁y₂ – λ₁x₁x₂? No… -λ₁y₁x₂ + λ₂y₂x₁) / det(P) = (-λ₁y₁x₂ + λ₂y₂x₁) / det(P)

Variables Table

Variables used in the find matrix from eigenvalues and eigenvectors calculator.

Variable	Meaning	Unit	Typical Range
λ1, λ2	Eigenvalues	Dimensionless	Real or complex numbers
x1, y1	Components of Eigenvector 1	Dimensionless	Real numbers
x2, y2	Components of Eigenvector 2	Dimensionless	Real numbers
P	Matrix of eigenvectors	Matrix	2×2 matrix of real numbers
D	Diagonal matrix of eigenvalues	Matrix	2×2 diagonal matrix
P^-1	Inverse of matrix P	Matrix	2×2 matrix, if det(P) ≠ 0
A	Original matrix	Matrix	2×2 matrix

Practical Examples (Real-World Use Cases)

Example 1:

Suppose we have eigenvalues λ₁ = 5, λ₂ = 1 and eigenvectors v₁ = [1, 1]^T, v₂ = [1, -1]^T.

P = [[1, 1], [1, -1]], D = [[5, 0], [0, 1]]

det(P) = (1)(-1) – (1)(1) = -2

P^-1 = (1/-2) * [[-1, -1], [-1, 1]] = [[0.5, 0.5], [0.5, -0.5]]

A = PDP^-1 = [[1, 1], [1, -1]] [[5, 0], [0, 1]] [[0.5, 0.5], [0.5, -0.5]] = [[5, 1], [5, -1]] [[0.5, 0.5], [0.5, -0.5]] = [[(5*0.5 + 1*0.5), (5*0.5 + 1*-0.5)], [(5*0.5 + -1*0.5), (5*0.5 + -1*-0.5)]] = [[3, 2], [2, 3]]

So, the original matrix is A = [[3, 2], [2, 3]]. Using the find matrix from eigenvalues and eigenvectors calculator with these inputs will yield this result.

Example 2:

Eigenvalues λ₁ = 2, λ₂ = -3 and eigenvectors v₁ = [1, 0]^T, v₂ = [1, 5]^T.

P = [[1, 1], [0, 5]], D = [[2, 0], [0, -3]]

det(P) = (1)(5) – (1)(0) = 5

P^-1 = (1/5) * [[5, -1], [0, 1]] = [[1, -0.2], [0, 0.2]]

A = PDP^-1 = [[1, 1], [0, 5]] [[2, 0], [0, -3]] [[1, -0.2], [0, 0.2]] = [[2, -3], [0, -15]] [[1, -0.2], [0, 0.2]] = [[(2*1 + -3*0), (2*-0.2 + -3*0.2)], [(0*1 + -15*0), (0*-0.2 + -15*0.2)]] = [[2, -1], [0, -3]]

The original matrix is A = [[2, -1], [0, -3]]. The find matrix from eigenvalues and eigenvectors calculator helps automate this.

How to Use This Find Matrix from Eigenvalues and Eigenvectors Calculator

Using the calculator is straightforward:

1. Enter Eigenvalue 1 (λ1): Input the first eigenvalue.
2. Enter Eigenvector 1 (v1x, v1y): Input the x and y components of the first eigenvector.
3. Enter Eigenvalue 2 (λ2): Input the second eigenvalue.
4. Enter Eigenvector 2 (v2x, v2y): Input the x and y components of the second eigenvector.
5. Calculate: The calculator automatically updates, or you can click "Calculate Matrix".
6. Read Results: The "Original Matrix A" is displayed prominently. Intermediate matrices P, D, P^-1, and det(P) are also shown. If det(P) is zero, an error will indicate the eigenvectors are linearly dependent.
7. Reset: Click "Reset" to clear inputs to default values.
8. Copy: Click "Copy Results" to copy the matrix A and intermediate values to your clipboard.

Ensure your eigenvectors are linearly independent (one is not a scalar multiple of the other) for a valid inverse P^-1 and a unique matrix A.

Key Factors That Affect Find Matrix from Eigenvalues and Eigenvectors Calculator Results

• Eigenvalues: The values of λ₁ and λ₂ directly determine the scaling along the eigenvector directions. Different eigenvalues lead to different matrices.
• Eigenvectors: The components of the eigenvectors define the directions along which the transformation A acts as simple scaling. They form the columns of P.
• Linear Independence of Eigenvectors: For a 2×2 matrix, if the two eigenvectors are linearly dependent (point in the same or exactly opposite directions), det(P) will be zero, P^-1 is undefined, and the matrix A cannot be reconstructed uniquely using this method unless it's a special case (like a scalar multiple of identity with repeated eigenvalues and any two independent vectors). The find matrix from eigenvalues and eigenvectors calculator will show an error if det(P) is 0.
• Order of Eigenvalues/Eigenvectors: If you swap λ₁ with λ₂ and v₁ with v₂, the resulting matrix A will be the same, but the intermediate matrices P and D will have their columns/diagonal elements swapped.
• Scaling of Eigenvectors: If you scale an eigenvector by a non-zero constant (e.g., use [2, 4] instead of [1, 2]), the matrix P changes, but P^-1 also changes in a way that the final matrix A remains the same. The find matrix from eigenvalues and eigenvectors calculator uses the exact components you enter.
• Numerical Precision: Very small or very large numbers, or det(P) very close to zero, might lead to precision issues in floating-point calculations, although this calculator uses standard JavaScript numbers.

Frequently Asked Questions (FAQ)

What if the eigenvalues are the same (repeated)?

If λ₁ = λ₂, and you still have two linearly independent eigenvectors, the matrix is a scalar multiple of the identity matrix (A = λI). If you have repeated eigenvalues but only one linearly independent eigenvector, the matrix is not diagonalizable, and this A=PDP^-1 method doesn't directly apply without modification (e.g., Jordan Normal Form).

What if the determinant of P is zero?

If det(P) = 0, the eigenvectors are linearly dependent. This means they lie on the same line through the origin. The matrix P is singular, and P^-1 does not exist. The original matrix A might not be uniquely determined or diagonalizable in the standard way. Our find matrix from eigenvalues and eigenvectors calculator flags this.

Can I use this calculator for 3×3 matrices?

No, this specific calculator is designed for 2×2 matrices only, as it has input fields for two eigenvalues and two 2-component eigenvectors. The principle extends to NxN matrices but requires N eigenvalues and N linearly independent N-component eigenvectors, and the matrix algebra is more complex.

What if the eigenvalues or eigenvectors are complex?

This calculator assumes real eigenvalues and eigenvectors. If they are complex, the matrix A can still be reconstructed using A=PDP^-1, but the calculations involve complex arithmetic, and the resulting matrix A might have complex entries (or real if eigenvalues/vectors come in conjugate pairs).

Is the reconstructed matrix A always unique?

If you have a full set of linearly independent eigenvectors for the given eigenvalues, and det(P) ≠ 0, then the matrix A reconstructed via A=PDP^-1 is unique.

How does the find matrix from eigenvalues and eigenvectors calculator handle non-numeric input?

It attempts to parse the input as numbers. If non-numeric input is provided, it will likely result in NaN (Not a Number) for the calculations, and the results will reflect that or show an error.

Why are eigenvalues and eigenvectors important?

They reveal fundamental properties of linear transformations represented by matrices. Eigenvectors are directions unchanged (only scaled) by the transformation, and eigenvalues are the scaling factors. They are used in many fields, like stability analysis, vibration analysis, and data analysis (PCA).

What does the chart show?

The chart visualizes the two input eigenvectors as vectors originating from (0,0) and ending at (v1x, v1y) and (v2x, v2y) respectively, helping to see their directions and relative magnitudes.

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