Eigenvector
An eigenvector is a nonzero vector that a linear transformation maps to a scalar multiple of itself.
Quantum Programming in Depth
An eigenstate is a quantum state that a quantum gate transforms into the same state multiplied by a scalar phase or eigenvalue.
- How it is used
- Supports phase kickback, unitary discrimination, and quantum phase estimation by supplying states whose transformations differ only by phase.
- How deep it goes
- Defines eigenvectors, eigenvalues, and eigenphases, then implements and tests two-bit phase estimation without deriving all algorithm variants.
- Examples
- Uses \(Z\), \(INC\), and \(T\) unitaries, \(\ket{0}\) and \(\ket{1}\), amplitude lists, preparation operations, and Qiskit and Q# tests.
- What you can do after
- Apply it via Qiskit or Q# by preparing a supplied eigenstate, running controlled-unitary phase estimation, and testing the returned eigenphase.
Read how Quantum Programming in Depth explains eigenvector →
Math and Architectures of Deep Learning
A nonzero vector that is scaled without changing direction by a linear transformation.
- How it is used
- Identifies invariant directions of linear transformations for diagonalization, geometric axes, and dimensionality reduction.
- How deep it goes
- Derives the characteristic condition, proves independence and orthogonality results, and applies diagonalization to ellipses and PCA.
- Examples
- Uses a Z-axis rotation, a 2D hyperellipse 5x² + 6xy + 5y² = 20, and PyTorch eigendecomposition.
- What you can do after
- Apply it via PyTorch to compute eigendecompositions and use eigenvector directions to align hyperellipse axes.
Read how Math and Architectures of Deep Learning explains eigenvector →
Data Without Labels
An eigenvector is a nonzero vector whose direction is unchanged when a linear transformation is applied, changing only by a scalar factor.
- How it is used
- Identifies variance-capturing directions used to select principal components for dimensionality reduction.
- How deep it goes
- Defines the matrix equation and works through covariance-based selection, while deferring broader mathematical treatment.
- Examples
- Uses Iris flower measurements, NumPy eigenvalue calculations, and scikit-learn PCA variance plots.
- What you can do after
- Apply it via NumPy and scikit-learn to inspect variance directions and choose PCA components.
Building Quantum Software with Python
An eigenstate, or eigenvector, is a nonzero state vector that a transformation maps to a scalar multiple of itself.
- How it is used
- Uses eigenstates to encode a circuit rotation phase for quantum phase estimation.
- How deep it goes
- Defines eigenstates through amplitude-wide phase rotation and verifies one with a single-qubit RY circuit.
- Examples
- Demonstrates the state [i/√2, 1/√2] under RY(π/3) and a target-register eigenstate in QPE.
- What you can do after
- Build it from scratch as an eigenstate-preparation step in a quantum phase estimation circuit.
Read how Building Quantum Software with Python explains eigenvector →
Related concepts
- diagonalizationused by · A matrix transformation technique that expresses a square matrix in a basis where it becomes diagonal, simplifying computations such as powers.
- quantum phase estimationused by · Quantum phase estimation is a quantum algorithm that estimates the eigenphase of a unitary operator given a corresponding eigenstate.
- principal component analysisused by · Principal component analysis is a linear dimensionality-reduction technique that transforms data into orthogonal components ordered by the variance they capture.
- eigenvalueuses · An eigenvalue is a scalar by which a linear transformation scales a corresponding nonzero eigenvector without changing its direction.
- orthogonalitycontrasts with · Orthogonality is the relationship between vectors, directions, or states whose inner product is zero, corresponding to perpendicularity in Euclidean space.
These guides are generated from the text of Manning books by Manning's knowledge graph and link to the sections they draw on. The full books are available on liveBook and at manning.com.