Eigenvalue
An eigenvalue is a scalar by which a linear transformation scales a corresponding nonzero eigenvector without changing its direction.
Data Without Labels
An eigenvalue is a scalar associated with a linear transformation and an eigenvector whose direction remains unchanged by that transformation.
- How it is used
- Ranks candidate principal components by the variance captured along their eigenvectors.
- How deep it goes
- Defines the eigenvalue–eigenvector relation and works through covariance-based selection, while deferring broader eigenvector mathematics.
- Examples
- Uses Iris flower measurements, covariance-matrix eigenvalues, explained-variance plots, and a reduction from four variables to two.
- What you can do after
- Apply it via NumPy and scikit-learn to inspect variance and choose principal components for dimensionality reduction.
Math and Architectures of Deep Learning
A scalar associated with a linear transformation that gives the scaling factor for a corresponding eigenvector.
- How it is used
- Uses eigenvalues to analyze linear transformations, identify data directions, diagonalize matrices, and reduce dimensionality.
- How deep it goes
- Derives the characteristic equation, proves independence and orthogonality properties, explains diagonalization, and demonstrates PyTorch computation.
- Examples
- Uses rotation axes, characteristic equations, symmetric matrices, covariance matrices, principal axes, and a PyTorch eigenvalue listing.
- What you can do after
- Evaluate eigenvalues with PyTorch and interpret their eigenvectors as scaling directions, rotation axes, or principal data axes.
Read how Math and Architectures of Deep Learning explains eigenvalue →
Building Quantum Software with Python
An eigenvalue is the scalar by which a transformation multiplies an associated eigenstate, and for a quantum circuit it is a complex number of magnitude one.
- How it is used
- Supports quantum phase estimation by encoding an eigenvalue phase as a measurable periodic frequency.
- How deep it goes
- Defines circuit eigenvalues, visualizes repeated phase rotation, and implements their use in QPE without deriving general linear algebra.
- Examples
- Uses an RY(2θ) single-qubit circuit and a three-qubit QPE estimate with θ = 4.7·2π/2³.
- What you can do after
- Build it from scratch in a QPE circuit by preparing an eigenstate, applying controlled circuit powers, and estimating its phase after IQFT.
Read how Building Quantum Software with Python explains eigenvalue →
Quantum Programming in Depth
An eigenvalue is the scalar, here a unit-modulus complex phase factor, by which a linear transformation scales an associated eigenvector.
- How it is used
- Frames phase estimation as finding the eigenphase associated with a supplied eigenvector and unitary operation.
- How deep it goes
- Defines eigenvalues, unit-modulus constraints, and eigenphases, then stops before detailing the phase-estimation procedure.
- Examples
- Uses Z, INC, and T gates with explicit eigenvectors, eigenvalues, decimal eigenphases, and binary eigenphases.
- What you can do after
- Evaluate/understand eigenvalues and convert unit-modulus eigenvalues into corresponding eigenphases.
Cyber Threat Hunting
An eigenvalue associated with a principal component quantifies the amount of variance that component captures from the data.
- How it is used
- Guides PCA feature reduction and helps select how many principal components to retain for analysis or modeling.
- How deep it goes
- Defines component strength, explains variance-based ordering, and stops short of deriving eigenvalues mathematically.
- Examples
- Uses 80% and 5% variance examples, PC1 versus the weakest component, and a four-component cumulative-variance plot.
- What you can do after
- Evaluate PCA components and select a retention count using eigenvalue strength and cumulative explained variance.
Related concepts
- quantum phase estimationmeasured by · Quantum phase estimation is a quantum algorithm that estimates the eigenphase of a unitary operator given a corresponding eigenstate.
- complex floating-point typekind of · A complex floating-point type represents complex numbers with real and imaginary components, using floating-point values for those components.
- spectral analysismeasured by · Spectral analysis examines a signal, time series, or system through the distribution and relationships of its spectral components, such as frequencies or eigenvalues.
- eigenvectorused by · An eigenvector is a nonzero vector that a linear transformation maps to a scalar multiple of itself.
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