Hey so I see you're looking for an online calculator for eigenvalues and eigenvectors. Yeah those are handy for a quick check on a homework problem or something but if you're doing anything serious you'll want to move past that pretty fast. I learned that the hard way a while back. I was working on this project for a small robotics firm we were building a simple vision system for a robotic arm to identify the orientation of an object on a conveyor belt. The core of it involved Principal Component Analysis or PCA which is basically a fancy way of finding the directions of highest variance in a point cloud. And what's at the heart of PCA? You guessed it finding the eigenvalues and eigenvectors of a covariance matrix. At first I was just grabbing sample data manually calculating the 3x3 covariance matrix and plugging it into some random website to get the principal components. It worked for the prototype. But then we needed to do it in real time for hundreds of frames per second. The whole 'copy-paste to a website' workflow fell apart obviously. I needed to do it programmatically. It forced me to actually understand what was going on instead of just getting an answer from a black box. The problem with doing it by hand for a 3x3 matrix is solving the characteristic equation `det(A - λI) = 0`. This gives you a cubic polynomial in λ. Finding the roots of a cubic equation is a massive pain and prone to errors. And then for each of those three eigenvalues λ you have to solve a system of linear equations `(A - λI)v = 0` to find the corresponding eigenvector v. It's doable but it's tedious and if you need precision you're gonna have a bad time. So let's ditch the online calculators and look at how you'd actually do this in a real-world scenario. The best way is to use a numerical library in a programming language you're comfortable with. The algorithms in these libraries like the QR algorithm are way more numerically stable and efficient than anything you'd do by hand. Here's how I'd approach it in Python which is what I ended up using for that robotics project. You'll need NumPy it's the standard for any numerical work in Python. ```python import numpy as np # Let's define our 3x3 matrix A # Make sure it's a NumPy array A = np.array([[4, 0, 1], [-2, 1, 0], [-2, 0, 1]]) print("Our Matrix A:\n", A) # This is the magic part. One function call. # eig() returns a tuple: a 1D array of eigenvalues # and a 2D array where each column is the corresponding eigenvector eigenvalues, eigenvectors = np.linalg.eig(A) print("\nEigenvalues (λ):") print(eigenvalues) print("\nEigenvectors (v) [as columns]:") print(eigenvectors) # Let's verify the first eigenvalue/eigenvector pair # A * v1 should be equal to λ1 * v1 lambda1 = eigenvalues[0] v1 = eigenvectors[:, 0] # First column is the first eigenvector # Calculate A * v1 Av1 = np.dot(A, v1) # Calculate λ1 * v1 lambda1_v1 = lambda1 * v1 print("\nVerification for the first pair:") print("A * v1 =", Av1) print("λ1 * v1 =", lambda1_v1) # The results should be very close (allowing for floating point inaccuracies) # np.allclose() is the right way to compare them print("Are they close? ->", np.allclose(Av1, lambda1_v1)) ``` See how simple that is? NumPy's `linalg.eig` function does all the heavy lifting. It's robust it's fast and it's reliable. The output gives you the eigenvalues and a matrix where the columns are the corresponding eigenvectors. I added a little verification step at the end which is super important. I remember one time the results were slightly off and it drove me crazy for a day until I realized it was just a floating point precision issue. My computer and I had a very stern talk about the IEEE 754 standard after that. You should always use a function like `np.allclose` to compare floats instead of `==`. Now maybe you're not a Python person. Maybe you're building something high-performance and need C++. You're in luck there too. The go-to library for this is called Eigen. Yes the library is named Eigen it's a little on the nose but it's an amazing header-only library for linear algebra in C++. Here's what that might look like. It's a bit more verbose than Python but you get that C++ speed. ```cpp #include #include int main() { // Define our 3x3 matrix A using the Eigen library Eigen::Matrix3f A; A << 4, 0, 1, -2, 1, 0, -2, 0, 1; std::cout << "Our Matrix A:\n" << A << std::endl; // Create an EigenSolver object for our matrix Eigen::EigenSolver solver(A); // The eigenvalues and eigenvectors can be complex numbers in general // so the solver returns them in a complex format Eigen::VectorXcf eigenvalues = solver.eigenvalues(); Eigen::MatrixXcf eigenvectors = solver.eigenvectors(); std::cout << "\nEigenvalues (λ):" << std::endl; // Eigenvalues can be complex, so we print them out std::cout << eigenvalues << std::endl; std::cout << "\nEigenvectors (v) [as columns]:" << std::endl; std::cout << eigenvectors << std::endl; // Verification for the first pair // Note: We need to use complex types for verification Eigen::MatrixXcf v1 = eigenvectors.col(0); std::complex lambda1 = eigenvalues(0); // Calculate A * v1 and λ1 * v1 // We need to cast our matrix A to complex to multiply Eigen::MatrixXcf Av1 = A.cast>() * v1; Eigen::MatrixXcf lambda1_v1 = lambda1 * v1; std::cout << "\nVerification for the first pair:" << std::endl; std::cout << "A * v1 =\n" << Av1 << std::endl; std::cout << "λ1 * v1 =\n" << lambda1_v1 << std::endl; // isApprox() is Eigen's version of np.allclose() bool is_close = Av1.isApprox(lambda1_v1); std::cout << "Are they close? -> " << (is_close ? "Yes" : "No") << std::endl; return 0; } ``` The C++ version with Eigen is a bit more involved. You have to think about data types like complex numbers because even a real matrix can have complex eigenvalues. But the core idea is the same: instantiate your matrix pass it to a solver and get your results. It's incredibly powerful for game development physics simulations or any place where performance is key. Since your original question was about an *online* calculator maybe you're interested in doing this in a web environment with JavaScript. You should not try to implement the QR algorithm from scratch in JS. Just don't. Use a library like `math.js`. It's a fantastic and extensive math library for both the browser and Node.js. Here's how you could use `math.js`. ```javascript // You'd typically install this with npm or include it via a script tag // For example: // This code assumes the math object is available globally // Define our 3x3 matrix A using math.js const A = math.matrix([ [4, 0, 1], [-2, 1, 0], [-2, 0, 1] ]); console.log("Our Matrix A:"); console.log(A.toArray()); // math.eigs() is the function you need // It returns an object with the values and vectors const result = math.eigs(A); // The result object contains 'values' and 'vectors' const eigenvalues = result.values; const eigenvectors = result.vectors; console.log("\nEigenvalues (λ):"); console.log(eigenvalues); console.log("\nEigenvectors (v) [as columns]:"); // The eigenvectors are stored as columns, let's print them nicely for (let i = 0; i < eigenvectors[0].length; i++) { const v = eigenvectors.map(row => row[i]); console.log(`Vector ${i+1}:`, v); } // Verification for the first pair const lambda1 = eigenvalues[0]; // Getting the first eigenvector (first column) const v1 = eigenvectors.map(row => row[0]); // A * v1 const Av1 = math.multiply(A, v1); // λ1 * v1 const lambda1_v1 = math.multiply(lambda1, v1); console.log("\nVerification for the first pair:"); console.log("A * v1 =", Av1.toArray()); console.log("λ1 * v1 =", lambda1_v1.toArray()); // math.deepEqual and math.abs can be used for a rough comparison const areClose = math.deepEqual(math.round(Av1, 5), math.round(lambda1_v1, 5)); console.log("Are they close (when rounded)? ->", areClose); ``` This JS example shows that no matter the ecosystem there's a tool for the job. `math.js` is great for interactive web applications or server-side calculations with Node.js. You could easily build your own online calculator with it. So instead of just finding one calculator you're now equipped to build your own or more importantly integrate this crucial calculation into a larger application. If you want to really understand the theory and the algorithms under the hood I can't recommend a couple of resources enough. Don't look for blog posts look for the foundational texts. First get a copy of "Linear Algebra and Its Applications" by Gilbert Strang. It's the bible for this stuff. He has an incredible way of giving you an intuitive feel for what eigenvalues and eigenvectors actually *are* in terms of stretching and rotation. Second if you want to understand the numerical methods themselves the algorithms that these libraries are actually running then you need to check out "Numerical Recipes" by Press Teukolsky Vetterling and Flannery. The C++ edition is a classic. It will walk you through the implementations of things like QR decomposition which is often used to find eigenvalues. It's dense but it's the real deal for anyone serious about scientific computing. So yeah online calculators are fine for a quick look but the moment you need to do something real repeatable or scalable you have to write code. Pick your language grab a solid linear algebra library and you're good to go. Hope this helps.

an online calculator for finding eigenvalues and eigenvectors of a 3x3 matrix