Linear Algebra and Finite-Dimensional Hilbert Spaces

Finite-dimensional Hilbert spaces are the cleanest laboratory for quantum mechanics. Qubits, spin multiplets, finite-level atoms, lattice sites, truncated bases, and many numerical models can all be described with complex vectors and matrices. In this setting every linear map is bounded, every operator is defined on the whole space, and completeness is automatic. The essential ideas of states, observables, unitary evolution, projection, composition, and spectral resolution can therefore be separated from the domain subtleties of infinite-dimensional analysis.

This chapter supplies that finite-dimensional mathematical spine. It explains the structure behind matrix calculations and routes each topic to one canonical page. Physical measurement rules belong in Core Formalism; physical composite systems and entanglement belong in Composite Systems and Entanglement; operator domains and continuous spectra belong in Infinite-Dimensional Hilbert Spaces.

The finite-dimensional model

An $n$-dimensional quantum state space is a complex inner-product space $\mathcal H$ isomorphic to $\mathbb C^n$. After choosing an orthonormal basis $\{|e_i\rangle\}_{i=1}^n$,

$$|\psi\rangle =\sum_{i=1}^n c_i|e_i\rangle, \qquad c_i=\langle e_i|\psi\rangle,$$

and the identity has the resolution

$$I=\sum_{i=1}^n |e_i\rangle\langle e_i|.$$

The basis converts vectors into columns and operators into matrices; it does not create the underlying objects. If a unitary matrix $S$ describes a change between orthonormal bases, then

$$c'=S^\dagger c, \qquad A'=S^\dagger A S.$$

Inner products, eigenvalues, traces, expectation values, and transition amplitudes remain invariant when all representations are transformed consistently. Begin with Finite-Dimensional Hilbert Spaces, Orthonormal Bases, and Change of Basis if this distinction is not yet automatic.

Four strands

The chapter is easiest to navigate as four connected strands.

1. Space and representation: finite-dimensional Hilbert spaces, orthonormal bases, and changes of basis establish the setting and its coordinate descriptions.
2. Operators and spectra: eigenvectors, diagonalization, Hermitian, unitary, and normal operators, projectors, and spectral decomposition organize observables and evolution.
3. Operator calculus and algebra: matrix functions, exponentials, commutators, and anticommutators turn spectral data into dynamics and structural identities.
4. Composition and factorization: tensor products, direct sums, singular value decomposition, and Schmidt decomposition distinguish ways of combining or decomposing spaces and maps.

Pauli matrices and Bloch-sphere geometry provide a two-dimensional meeting point where all four strands can be seen explicitly.

Operator classes and the spectral theorem

Several operator classes recur throughout quantum mechanics. Their definitions are short, but their implications differ.

Class	Defining condition	Finite-dimensional consequence	Typical role
Hermitian	$A=A^\dagger$	real eigenvalues and an orthonormal eigenbasis	observables and generators
Unitary	$U^\dagger U=UU^\dagger=I$	norm preservation and eigenvalues of unit modulus	basis changes, symmetries, evolution
Normal	$NN^\dagger=N^\dagger N$	unitary diagonalizability	common spectral framework
Orthogonal projector	$P=P^\dagger=P^2$	projection onto a subspace	alternatives and spectral subspaces

Hermitian and unitary operators are normal, but a normal operator need be neither Hermitian nor unitary. The finite-dimensional spectral theorem states that a normal operator has a decomposition

$$N=\sum_{\lambda}\lambda P_{\lambda}, \qquad I=\sum_{\lambda}P_{\lambda}, \qquad P_{\lambda}P_{\mu} =\delta_{\lambda\mu}P_{\lambda}.$$

Each $P_\lambda$ projects onto the full eigenspace for $\lambda$, so this form handles degeneracy without selecting a preferred basis inside a degenerate subspace. Hermiticity constrains $\lambda$ to be real; unitarity constrains $\lvert\lambda\rvert=1$.

The same projectors define functions of the operator:

$$f(N)=\sum_{\lambda}f(\lambda)P_{\lambda}.$$

This formula links Spectral Decomposition to Matrix Functions and Exponentials. It is the finite-dimensional prototype for the spectral measures and functional calculus used later in infinite-dimensional quantum mechanics.

Eigenvectors are not the whole story

The equation $Av=\lambda v$ identifies directions preserved by an operator. It does not guarantee that eigenvectors span the space. A matrix is diagonalizable exactly when it has enough linearly independent eigenvectors; normality is a stronger condition that guarantees an orthonormal eigenbasis.

This hierarchy matters:

$$\text{Hermitian or unitary} \Longrightarrow \text{normal} \Longrightarrow \text{unitarily diagonalizable} \Longrightarrow \text{diagonalizable}.$$

The reverse implications generally fail. Use Eigenvalues and Eigenvectors for the eigenvalue equation and degeneracy, Diagonalization for the spanning question, and Normal Operators for the exact criterion for unitary diagonalization.

Commutators and simultaneous structure

The commutator and anticommutator are

$$[A,B]=AB-BA, \qquad \{A,B\}=AB+BA.$$

For Hermitian operators in finite dimension, $[A,B]=0$ is equivalent to the existence of a common orthonormal eigenbasis. Degeneracy must be handled at the level of invariant eigenspaces: commuting with $A$ means that $B$ preserves each eigenspace of $A$, after which $B$ can be diagonalized within those subspaces.

Anticommutators encode a different algebraic relation and are especially useful for Pauli matrices and fermionic systems. The dedicated Commutators and Anticommutators page owns the identities; the physical relation among compatibility, measurement statistics, and uncertainty is developed in Compatibility, Commutators, and Uncertainty.

Direct sums and tensor products

Direct sums and tensor products combine spaces in fundamentally different ways.

Construction	Dimension	Basic vectors	Interpretation
$V\oplus W$	$\dim V+\dim W$	ordered pairs $(v,w)$	sectors, alternatives, or invariant blocks
$V\otimes W$	$(\dim V)(\dim W)$	linear combinations of $v\otimes w$	joint degrees of freedom

A block-diagonal operator naturally acts on a direct sum. A local operator $A\otimes I$ naturally acts on one factor of a tensor product. Replacing one construction by the other changes both dimension and physical meaning.

The mathematical construction of product spaces and product operators is canonical in Tensor Products. The interpretation of those spaces as physical composites, together with reduced states and entanglement, belongs in Composite Systems and Entanglement Basics and the dedicated composite-systems volume.

Eigensystems, SVD, and Schmidt decomposition

Three decompositions answer different questions.

Decomposition	Applies to	Canonical form	What it reveals
Spectral decomposition	normal endomorphism $N:V\to V$	$N=U\Lambda U^\dagger$	invariant eigenmodes and eigenvalues
Singular value decomposition	arbitrary map $T:V\to W$	$T=U\Sigma V^\dagger$	rank, input-output directions, and nonnegative stretches
Schmidt decomposition	bipartite vector in $V\otimes W$	$\lvert\psi\rangle=\sum_r s_r\lvert u_r\rangle\lvert v_r\rangle$	bipartite correlation structure

The SVD exists even for rectangular maps and nondiagonalizable matrices. Applying it to the coefficient matrix of a bipartite vector gives the Schmidt decomposition. The singular values become Schmidt coefficients, and normalization gives $\sum_r s_r^2=1$. The mathematical derivation belongs here; entanglement measures and operational consequences belong in the composite-systems volume.

A two-level spectral calculation

Every Hermitian two-by-two matrix can be written

$$H=h_0I+\mathbf h\cdot\boldsymbol\sigma, \qquad h_0\in\mathbb R, \quad \mathbf h\in\mathbb R^3.$$

For $\mathbf h\ne0$, define $\hat{\mathbf h}=\mathbf h/\lVert\mathbf h\rVert$. Since $(\hat{\mathbf h}\cdot\boldsymbol\sigma)^2=I$, the eigenvalues and projectors are

$$E_\pm=h_0\pm\lVert\mathbf h\rVert, \qquad P_\pm=\frac12 \left(I\pm\hat{\mathbf h}\cdot\boldsymbol\sigma\right).$$

Consequently,

$$H=E_+P_++E_-P_-,$$

and functional calculus gives the unitary evolution directly:

$$e^{-itH/\hbar} =e^{-itE_+/\hbar}P_+ +e^{-itE_-/\hbar}P_-.$$

This one calculation connects Hermitian Operators, Projectors, Spectral Decomposition, and Matrix Functions and Exponentials. The local Pauli Matrices page is an algebraic bridge; their canonical physical role as spin operators is developed in Pauli Matrices for Spin.

Page map

Page	Use it to answer…
Finite-Dimensional Hilbert Spaces	What extra structure turns $\mathbb C^n$ into a quantum state space?
Orthonormal Bases	How are coefficients extracted, and what does completeness mean here?
Change of Basis	How do vectors and operators transform without changing predictions?
Eigenvalues and Eigenvectors	Which directions does an operator preserve, and how is degeneracy defined?
Diagonalization	When does an eigenbasis exist?
Hermitian Operators	Why are observable matrices associated with real spectra and orthogonal eigenspaces?
Unitary Operators	Which maps preserve inner products and norms?
Normal Operators	Which matrices are unitarily diagonalizable?
Projectors	How are subspaces isolated algebraically?
Spectral Decomposition	How is an operator reconstructed from eigenvalues and eigenspace projectors?
Matrix Functions and Exponentials	How are $f(A)$ and $e^A$ computed and interpreted?
Commutators and Anticommutators	What algebraic information is carried by $AB-BA$ and $AB+BA$?
Tensor Products	How are joint vector spaces and product operators constructed?
Direct Sums	How are independent sectors and block structures assembled?
Singular Value Decomposition	How does an arbitrary map split into orthogonal directions and stretches?
Schmidt Decomposition as Linear Algebra	Why is a bipartite pure-state decomposition an SVD?
Pauli Matrices	Which identities organize two-by-two matrix algebra?
Bloch Sphere Geometry	How do one-qubit states and unitary rotations become Euclidean geometry?

Suggested routes

• Core formalism: spaces $\to$ bases $\to$ Hermitian operators $\to$ projectors $\to$ spectral decomposition $\to$ unitary operators.
• Spin and two-level systems: spaces $\to$ Hermitian and unitary operators $\to$ Pauli matrices $\to$ matrix exponentials $\to$ Bloch-sphere geometry.
• Composite systems: tensor products $\to$ direct sums $\to$ SVD $\to$ Schmidt decomposition.
• Numerical work: change of basis $\to$ diagonalization $\to$ SVD, followed by Matrix Diagonalization and Conditioning and Stability.
• Infinite-dimensional quantum mechanics: spectral decomposition $\to$ Hilbert Spaces $\to$ Domains of Operators $\to$ Spectral Theorem, Practical Version.

Common mistakes

Mistake	Correction
Treating every matrix as diagonalizable	check whether a full eigenbasis exists; use the SVD when appropriate
Assuming real eigenvalues imply Hermiticity	Hermiticity is sufficient, not necessary, for a real spectrum
Confusing Hermitian with real symmetric	complex Hermitian matrices may contain nonreal off-diagonal entries
Replacing normality by Hermiticity	Hermitian and unitary operators are distinct subclasses of normal operators
Choosing eigenvectors inside a degenerate eigenspace as though they were unique	use the eigenspace projector for basis-independent statements
Confusing $V\oplus W$ with $V\otimes W$	compare dimensions and identify whether the construction represents sectors or joint degrees of freedom
Applying eigendecomposition to a rectangular coefficient matrix	use the SVD; its singular values produce Schmidt coefficients
Extending finite-dimensional operator identities without checking domains	move to the infinite-dimensional pages and state domains explicitly

Exercises

1. Spectral data and operator class

Suppose $N=\sum_r\lambda_rP_r$, where the $P_r$ are mutually orthogonal projectors resolving the identity. Show that $N$ is normal. State the additional condition on the $\lambda_r$ for $N$ to be Hermitian or unitary.

Solution

The adjoint is $N^\dagger=\sum_r\lambda_r^*P_r$. Orthogonality gives

$$NN^\dagger =\sum_r\lvert\lambda_r\rvert^2P_r =N^\dagger N,$$

so $N$ is normal. It is Hermitian exactly when every $\lambda_r$ is real. It is unitary exactly when every $\lvert\lambda_r\rvert=1$ on the resolved space.

2. Direct sum or tensor product?

Let $V\cong\mathbb C^2$ and $W\cong\mathbb C^3$. Find the dimensions of $V\oplus W$ and $V\otimes W$. Which construction describes a system with a two-level subsystem and a three-level subsystem present jointly?

Solution

The dimensions are

$$\dim(V\oplus W)=2+3=5, \qquad \dim(V\otimes W)=2\cdot3=6.$$

Joint subsystems use $V\otimes W$. The direct sum instead describes alternatives or sectors whose dimensions add.

3. Functional calculus from projectors

Let $A=\sum_r a_rP_r$ be Hermitian. Prove that $A^k=\sum_r a_r^kP_r$ for every nonnegative integer $k$, and use a convergent power series to justify $f(A)=\sum_r f(a_r)P_r$.

Solution

Because $P_rP_s=\delta_{rs}P_r$, multiplying two spectral sums keeps only equal-index terms. Induction gives

$$A^k=\sum_r a_r^kP_r.$$

If $f(z)=\sum_{k=0}^\infty c_kz^k$ converges on the finite spectrum, then

$$f(A) =\sum_k c_kA^k =\sum_r\left(\sum_kc_ka_r^k\right)P_r =\sum_r f(a_r)P_r.$$

Only finitely many spectral projectors occur, so exchanging the finite projector sum with the convergent scalar series is immediate.

4. From SVD to Schmidt coefficients

Write a normalized bipartite vector as $|\psi\rangle=\sum_{ij}C_{ij}|i\rangle|j\rangle$. If $C=U\Sigma V^\dagger$, explain why the singular values $s_r$ are Schmidt coefficients and show that $\sum_rs_r^2=1$.

Solution

Substituting the SVD and absorbing the columns of $U$ and the complex-conjugated columns of $V$ into new orthonormal local bases gives

$$|\psi\rangle =\sum_r s_r|u_r\rangle|v_r\rangle.$$

This is the Schmidt form. Normalization is the Frobenius norm of the coefficient matrix:

$$1=\langle\psi|\psi\rangle =\operatorname{tr}(C^\dagger C) =\sum_r s_r^2.$$

References

• S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2012.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.