Diagonalization | KEVOS® Mathematics

Q: What is the difference between diagonalisable and invertible?

They are independent properties. The zero matrix is diagonal, hence diagonalisable, but not invertible; the shear $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ is invertible but not diagonalisable. Invertibility asks whether zero is an eigenvalue; diagonalisability asks whether the eigenvectors span the space.

Q: Does a repeated eigenvalue prevent diagonalisation?

No. It only means the question must be answered by computation rather than by inspection. If the eigenspace of the repeated eigenvalue has dimension equal to its algebraic multiplicity, the matrix is diagonalisable; the identity matrix has one eigenvalue of multiplicity $n$ and is already diagonal.

Q: Is the diagonalisation of a matrix unique?

No. The columns of $S$ can be reordered, which permutes the diagonal of $D$, and each column can be rescaled by any non-zero scalar. Within an eigenspace of dimension greater than one, any basis at all may be used. What is unique is the multiset of diagonal entries, since those are the eigenvalues with their multiplicities.

Q: How does diagonalisation relate to the singular value decomposition?

They answer different questions. Diagonalisation uses a similarity transformation $S^{-1}AS$, preserves eigenvalues, applies only to square matrices and may not exist. The SVD uses two different unitary matrices, $U^{*}AV$, applies to any rectangular matrix, always exists and reveals magnitudes rather than invariant directions. They coincide, up to signs, for a symmetric positive definite matrix.

Q: What happens when a matrix cannot be diagonalised?

The closest attainable form under similarity is the Jordan form: block diagonal with each block having one eigenvalue on the diagonal and ones on the superdiagonal. It is exact but extremely sensitive to perturbation, so numerical work uses the Schur decomposition $A = QTQ^{*}$ with $Q$ unitary and $T$ upper triangular instead, which always exists and is computed stably.

Q: Why is $A^{k} = SD^{k}S^{-1}$ so much cheaper than repeated multiplication?

Because the diagonal of $D^{k}$ is just $\lambda_i^{k}$, so the entire power is $n$ scalar exponentiations plus two matrix products, independent of $k$. Repeated squaring of $A$ costs $O(n^3 \log k)$ operations, and naive repeated multiplication $O(n^3 k)$. The saving grows with $k$, and the same substitution extends to $e^{At}$ and to any analytic function of the matrix.

Q: Can I tell a matrix is diagonalisable without computing eigenvectors?

Sometimes. A Hermitian or real symmetric matrix always is; so is any matrix with $n$ distinct eigenvalues, which can be confirmed from the characteristic polynomial alone; and more generally any normal matrix, satisfying $A^{*}A = AA^{*}$, is unitarily diagonalisable. Outside these classes, a repeated eigenvalue forces an eigenspace dimension computation.

Overview

Similarity lets a matrix be rewritten in a different basis without disturbing its eigenvalues, its determinant, its rank or its characteristic polynomial. The natural question is how simple a form can be reached. Diagonalization answers it for the best case: a matrix $A$ is diagonalisable when some invertible $S$ makes $S^{-1}AS$ diagonal, at which point every off-diagonal coupling has been removed and the remaining diagonal entries are precisely the eigenvalues of $A$ .

The characterisation is constructive rather than existential. $A$ of size $n$ is diagonalisable if and only if $\mathbb{C}^n$ has a basis made entirely of eigenvectors of $A$ . Placing those eigenvectors in the columns of $S$ , in any order, produces the diagonalising matrix directly, and the eigenvalues appear on the diagonal in that same order. There is no search and no iteration — the eigenvectors are the answer.

Whether such a basis exists is decided by a count. Each eigenvalue $\lambda$ carries an algebraic multiplicity $\alpha_A(\lambda)$ , its multiplicity as a root of the characteristic polynomial, and a geometric multiplicity $\gamma_A(\lambda)$ , the dimension of its eigenspace. The geometric multiplicity never exceeds the algebraic one. Diagonalisability holds exactly when the two agree for every eigenvalue, because only then do the eigenspaces contribute enough independent directions to fill the whole space.

The engineering value is that a diagonal matrix acts independently on each coordinate. Powers, exponentials, inverses and polynomials of a diagonalisable matrix reduce to the same operation applied to $n$ scalars. A coupled linear recurrence or a coupled system of ordinary differential equations becomes $n$ uncoupled scalar problems, solved in closed form, then recombined. That decoupling is the whole reason modal analysis, principal component analysis and stability classification of linear systems are practical.

Definition

Diagonal Matrix

DIM

A square matrix $A$ is diagonal when every entry off the main diagonal is zero, that is $[A]_{ij} = 0$ whenever $i \ne j$ . The diagonal entries themselves are unrestricted and may be zero. Writing $D = \operatorname{diag}(d_1, d_2, \ldots, d_n)$ fixes the whole matrix with $n$ numbers instead of $n^2$ .

A diagonal matrix is simultaneously upper triangular and lower triangular, so its eigenvalues are its diagonal entries and its determinant is their product.

Diagonalisable Matrix

DZM

A square matrix $A$ is diagonalisable when it is similar to a diagonal matrix — that is, when there exists a non-singular matrix $S$ of the same size and a diagonal matrix $D$ with $S^{-1}AS = D$ . The pair $(S, D)$ is a diagonalisation of $A$ ; it is never unique, since reordering the columns of $S$ permutes the diagonal of $D$ correspondingly.

Diagonalisability is a property of the matrix, not of any particular $S$ . Because similarity is an equivalence relation, every matrix similar to a diagonalisable matrix is itself diagonalisable.

Defective Matrix

DFM

A square matrix is defective when at least one eigenvalue $\lambda$ satisfies $\gamma_A(\lambda) < \alpha_A(\lambda)$ , so that eigenvalue supplies fewer independent eigenvectors than its algebraic multiplicity. A matrix is diagonalisable precisely when it is not defective. The shortfall $\alpha_A(\lambda) - \gamma_A(\lambda)$ is the defect at $\lambda$ , and it is exactly the number of generalised eigenvectors that must be recruited to complete a basis.

Concepts

The eigenvector-basis characterisation

The central theorem states that a square matrix $A$ of size $n$ is diagonalisable if and only if there exists a linearly independent set of $n$ eigenvectors of $A$ . The reverse direction is the construction: given independent eigenvectors $\vec{x}_1, \ldots, \vec{x}_n$ for eigenvalues $\lambda_1, \ldots, \lambda_n$ , form $S = [\, \vec{x}_1 \mid \vec{x}_2 \mid \cdots \mid \vec{x}_n \,]$ . Then $A S$ has columns $A\vec{x}_i = \lambda_i \vec{x}_i$ , which is precisely $S$ with its $i$ -th column scaled by $\lambda_i$ , so $AS = SD$ . Independence makes $S$ non-singular, and multiplying on the left by $S^{-1}$ gives $S^{-1}AS = D$ . The proof is therefore an algorithm: locate $n$ independent eigenvectors and the diagonalisation writes itself.

The forward direction reads eigenvectors off the columns

The converse argument runs the same computation backwards. Suppose $T^{-1}AT = E$ with $E$ diagonal and $T$ non-singular. Then $AT = TE$ , and comparing columns gives $A\vec{y}_i = d_i \vec{y}_i$ for each column $\vec{y}_i$ of $T$ . Because $T$ is non-singular no column is the zero vector, so each column genuinely is an eigenvector, and the columns of a non-singular matrix form a linearly independent set. A diagonalisation therefore is an eigenvector basis, written in matrix form. Nothing else can diagonalise a matrix.

The multiplicity criterion

Counting eigenvectors directly is awkward, so the practical test is stated in multiplicities: $A$ is diagonalisable if and only if $\gamma_A(\lambda) = \alpha_A(\lambda)$ for every eigenvalue $\lambda$ of $A$ . The forward implication uses the fact that algebraic multiplicities sum to $n$ while the eigenvectors drawn from each eigenspace can never exceed its dimension; if one eigenvalue is deficient the total falls short of $n$ and no basis of eigenvectors exists. The reverse implication needs the theorem that eigenvectors belonging to distinct eigenvalues are linearly independent, which lets bases of the individual eigenspaces be concatenated into one independent set of size $\sum \gamma_A(\lambda_i) = \sum \alpha_A(\lambda_i) = n$ .

Distinct eigenvalues force diagonalisability

If a matrix of size $n$ has $n$ distinct eigenvalues then every algebraic multiplicity equals $1$ . Since $1 \le \gamma_A(\lambda) \le \alpha_A(\lambda)$ always holds, every geometric multiplicity is forced to $1$ as well, the multiplicity criterion is satisfied automatically, and the matrix is diagonalisable. This is a sufficient condition only. Plenty of diagonalisable matrices have repeated eigenvalues — the identity matrix is the extreme case, already diagonal with one eigenvalue of multiplicity $n$ — so a repeated root is a prompt to compute an eigenspace dimension, not a verdict.

Powers, polynomials and functions of a matrix

Write $A = SDS^{-1}$ . Then $A^2 = SDS^{-1}SDS^{-1} = SD^2S^{-1}$ , and by induction $A^{k} = SD^{k}S^{-1}$ for every non-negative integer $k$ . Because $D^{k}$ is diagonal with entries $\lambda_i^{k}$ , a power that would otherwise cost $k-1$ matrix products collapses to $n$ scalar powers plus two matrix products. The same substitution defines any function of the matrix: for a polynomial $q$ , $q(A) = S\, q(D)\, S^{-1}$ with $q$ applied entrywise on the diagonal, and the matrix exponential $e^{At} = S \operatorname{diag}(e^{\lambda_1 t}, \ldots, e^{\lambda_n t}) S^{-1}$ follows by the same route. This is where diagonalisation earns its keep in dynamics.

Not every matrix is diagonalisable

Consider $B$ with characteristic polynomial $p_B(x) = (x-1)(x-2)^3$ whose eigenspace for $\lambda = 2$ has dimension $1$ . Then $\gamma_B(2) = 1$ while $\alpha_B(2) = 3$ , the multiplicity criterion fails, and $B$ is not similar to any diagonal matrix. Failure is not exotic: the simplest instance is the $2 \times 2$ shear $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ , which has the single eigenvalue $1$ with $\alpha = 2$ and $\gamma = 1$ . For defective matrices the best available canonical form under similarity is the Jordan form, which is block diagonal with nilpotent superdiagonal blocks rather than fully diagonal.

The spectral projector view

A diagonalisation can be rewritten without any inverse in sight. Let $\vec{x}_i$ be the columns of $S$ and $\vec{w}_i^{\,t}$ the rows of $S^{-1}$ ; then $A = \sum_{i=1}^{n} \lambda_i \vec{x}_i \vec{w}_i^{\,t}$ , a sum of rank-one terms. Grouping the terms belonging to a single eigenvalue gives a projector onto that eigenspace along the others, and $A$ becomes a weighted sum of projectors. This form makes the modal interpretation explicit: applying $A$ decomposes a vector into its components in each eigenspace, scales each by its eigenvalue, and reassembles.

Procedure: decide and construct a diagonalisation

Form the characteristic polynomialCompute

p_A(x) = \det(A - xI_n)

and factor it. Record each distinct eigenvalue

\lambda_i

with its algebraic multiplicity

\alpha_A(\lambda_i)

Compute each eigenspaceFor each

\lambda_i

, row-reduce

A - \lambda_i I_n

and read a basis for its null space. The number of basis vectors is the geometric multiplicity

\gamma_A(\lambda_i)

Compare the multiplicitiesIf

\gamma_A(\lambda_i) < \alpha_A(\lambda_i)

for any

i

, the matrix is defective. Stop: no diagonalisation exists, and a Jordan form is the appropriate target instead.

Assemble $S$ Concatenate the eigenspace bases into a single list of

n

vectors and place them as the columns of

S

. Independence is guaranteed by the multiplicity test, so

S

is non-singular.

Write $D$ by inspectionSet

D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)

with each eigenvalue repeated as often as its column appears in

S

, in exactly the same order. No further computation is needed.

VerifyConfirm

AS = SD

by direct multiplication. This avoids inverting

S

and catches transcription errors that a check of

S^{-1}AS

would hide behind extra arithmetic.

Equations

Definition of diagonalisability

EQ-DZ-01

S^{- 1} A S = D = diag (λ_{1}, λ_{2}, \dots, λ_{n})

The matrix $S$ is non-singular and $D$ is diagonal. The diagonal entries of $D$ are the eigenvalues of $A$ , repeated according to their algebraic multiplicities.

Construction of the diagonalising matrix

EQ-DZ-02

S = [{\vec{x}}_{1} ∣ {\vec{x}}_{2} ∣ \dots ∣ {\vec{x}}_{n}], A {\vec{x}}_{i} = λ_{i} {\vec{x}}_{i}

Any linearly independent set of $n$ eigenvectors, taken in any order, gives a valid $S$ . The order chosen for the columns fixes the order of the eigenvalues on the diagonal of $D$ .

Column form of the diagonalisation

EQ-DZ-03

A S = S D

The inverse-free restatement of $S^{-1}AS = D$ . Reading it column by column recovers $A\vec{x}_i = \lambda_i\vec{x}_i$ , which is why the theorem is an eigenvector statement in disguise. This is also the cheapest way to verify a computed diagonalisation.

Multiplicity criterion for diagonalisability

EQ-DZ-04

A diagonalisable \Leftrightarrow γ_{A} (λ) = α_{A} (λ) \forall λ \in σ (A)

Here $\sigma(A)$ is the set of distinct eigenvalues, $\alpha_A$ the algebraic multiplicity and $\gamma_A$ the geometric multiplicity. Since $1 \le \gamma_A(\lambda) \le \alpha_A(\lambda)$ always, only a strict shortfall can break the criterion.

Eigenspace dimensions sum to the matrix size

EQ-DZ-05

\sum_{i}^{=} γ_{A} (λ_{i}) = n

An equivalent phrasing of the criterion for a matrix with $k$ distinct eigenvalues. Because algebraic multiplicities always sum to $n$ , equality here forces $\gamma_A = \alpha_A$ term by term.

Powers of a diagonalisable matrix

EQ-DZ-06

A^{k} = S D^{k} S^{- 1}, D^{k} = diag (λ_{1}^{k}, λ_{2}^{k}, \dots, λ_{n}^{k})

Reduces $k-1$ matrix multiplications to $n$ scalar powers and two matrix products. The same identity holds for negative $k$ when every eigenvalue is non-zero, and extends to any function applied to the diagonal.

Spectral expansion into rank-one terms

EQ-DZ-07

Spectral expansion into rank-one terms

The outer-product form of a diagonalisation. Each term projects onto one eigendirection and scales it, which is the algebraic statement of modal decomposition.

Distinct eigenvalues are sufficient

EQ-DZ-08

| σ (A) | = n \Rightarrow A diagonalisable

A matrix of size $n$ with $n$ distinct eigenvalues is always diagonalisable. The implication does not reverse: the identity matrix is diagonalisable with a single eigenvalue.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$A$	Subject matrix	The square matrix being tested for diagonalisability	n x n complex matrix
$S$	Diagonalising matrix	Non-singular matrix whose columns are $n$ independent eigenvectors of $A$	n x n non-singular matrix
$D$	Diagonal form	The diagonal matrix similar to $A$ , carrying the eigenvalues on its diagonal	n x n diagonal matrix
$λ_{i}$	Eigenvalue	A root of the characteristic polynomial of $A$	complex scalar
${\vec{x}}_{i}$	Eigenvector	A non-zero vector with $A\vec{x}_i = \lambda_i \vec{x}_i$ , used as a column of $S$	non-zero vector in C^n
$α_{A} (λ)$	Algebraic multiplicity	Multiplicity of $\lambda$ as a root of $p_A(x)$	integer 1 to n
$γ_{A} (λ)$	Geometric multiplicity	Dimension of the eigenspace $\mathcal{E}_A(\lambda) = \mathcal{N}(A - \lambda I_n)$	integer 1 to alpha
$p_{A} (x)$	Characteristic polynomial	$\det(A - xI_n)$ , whose roots are the eigenvalues	degree-n polynomial
$k$	Power index	Exponent applied to $A$ when using the diagonalisation to compute $A^{k}$	non-negative integer

Worked Numerical Example

Problem statement

A three-buffer production line is modelled by the one-period update $\vec{s}_{k+1} = A\vec{s}_k$ , where $A$ is the gain matrix below. Decide whether $A$ is diagonalisable, construct an explicit diagonalisation, and use it to obtain the ten-period propagator $A^{10}$ in closed form.

State the gain matrix
The three buffer levels are coupled by the following one-period update matrix. Note that it is not symmetric, so nothing about its spectrum is guaranteed in advance.
$A = [\begin{matrix} 7 & - 2 & - 2 \\ 6 & - 1 & - 2 \\ 6 & - 2 & - 1 \end{matrix}]$
Find the eigenvalues
The trace is $7 - 1 - 1 = 5$ , the sum of the three principal $2\times 2$ minors is $-3 + 5 + 5 = 7$ , and $\det(A) = 3$ . These are the coefficients of the characteristic polynomial up to sign, giving a factorisation with a repeated root.
$p_{A} (x) = x^{3} - 5 x^{2} + 7 x - 3 = (x - 3) (x - 1)^{2}$
Record the algebraic multiplicities
The distinct eigenvalues are $\lambda = 3$ and $\lambda = 1$ , with $\alpha_A(3) = 1$ and $\alpha_A(1) = 2$ . Because $\lambda = 1$ is repeated, diagonalisability is genuinely in question and must be settled by computing the eigenspace dimension.
Compute the eigenspace for $\lambda = 3$
Row-reduce $A - 3I_3$ . The reduced form has two pivot columns, so the null space has dimension $1$ and $\gamma_A(3) = 1$ , matching $\alpha_A(3) = 1$ as the general bound already guarantees.
$Compute the eigenspace for $\lambda = 3$$
Compute the eigenspace for $\lambda = 1$
Here $A - I_3$ has three identical rows, so its reduced form has a single pivot. Two free variables give a two-dimensional null space, hence $\gamma_A(1) = 2 = \alpha_A(1)$ . The multiplicity criterion is satisfied for both eigenvalues, so $A$ is diagonalisable.
$Compute the eigenspace for $\lambda = 1$$
Assemble $S$ and $D$
Place the three eigenvectors as columns in the order $3, 1, 1$ and write the eigenvalues on the diagonal in the same order. The inverse is computed once by reducing $[\, S \mid I_3 \,]$ .
$S = [\begin{matrix} 1 & 1 & 1 \\ 1 & 3 & 0 \\ 1 & 0 & 3 \end{matrix}], D = [\begin{matrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], S^{- 1} = [\begin{matrix} 3 & - 1 & - 1 \\ - 1 & \frac{2}{3} & \frac{1}{3} \\ - 1 & \frac{1}{3} & \frac{2}{3} \end{matrix}]$
Verify without inverting
Check $AS = SD$ column by column. Column one: $A(1,1,1)^{t} = (3,3,3)^{t} = 3(1,1,1)^{t}$ . Column two: $A(1,3,0)^{t} = (1,3,0)^{t}$ . Column three: $A(1,0,3)^{t} = (1,0,3)^{t}$ . All three match the corresponding columns of $S D$ , confirming the diagonalisation.
$A S = [\begin{matrix} 3 & 1 & 1 \\ 3 & 3 & 0 \\ 3 & 0 & 3 \end{matrix}] = S D$
Build the ten-period propagator
Since $D^{k} = \operatorname{diag}(3^{k}, 1, 1) = I_3 + (3^{k} - 1)\,\vec{e}_1\vec{e}_1^{\,t}$ , the power collapses to the identity plus a single rank-one correction: $A^{k} = I_3 + (3^{k} - 1)\,\vec{v}\vec{w}^{\,t}$ where $\vec{v} = (1,1,1)^{t}$ is the first column of $S$ and $\vec{w}^{\,t} = (3,-1,-1)$ is the first row of $S^{-1}$ . At $k = 10$ , $3^{10} - 1 = 59048$ .
$Build the ten-period propagator$

Result

The line has one growing mode with gain $3$ per period along the direction $(1,1,1)^{t}$ and a two-dimensional subspace on which the state is exactly preserved. Ten periods of propagation, which would otherwise need nine matrix products, reduce to one scalar power and a rank-one update. Physically, any initial imbalance lying in the $\lambda = 1$ plane persists indefinitely without growth, while any component along $(1,1,1)^{t}$ amplifies by a factor of $59049$ over ten periods — the design signal that this configuration is unstable and needs feedback.

Applications & Industry Use

Mechanical & structural dynamics

Modal analysis of vibrating structures

The generalised eigenproblem for a mass and stiffness pair diagonalises the equations of motion, replacing a coupled multi-degree-of-freedom system with independent single-degree-of-freedom oscillators. Each mode has its own natural frequency and can be truncated or damped separately, which is why modal superposition dominates earthquake and rotor-dynamic analysis.

Control systems

Decoupling state-space models

For $\frac{d\vec{x}}{dt} = A\vec{x} + B\vec{u}$ with $A$ diagonalisable, the change of coordinates $\vec{z} = S^{-1}\vec{x}$ produces $n$ independent scalar dynamics with rates $\lambda_i$ . Stability, settling time and the presence of oscillatory modes are then read off the diagonal, and controllability of an individual mode can be assessed in isolation.

Chemical & process engineering

Multi-component reaction kinetics

A first-order reaction network gives $\frac{d\vec{c}}{dt} = K\vec{c}$ for a rate matrix $K$ . Diagonalising $K$ yields reaction modes decaying as $e^{\lambda_i t}$ , exposing the slow mode that governs residence time and the fast modes that can be treated as instantaneous in a reduced model.

Data science & statistics

Principal component analysis

A sample covariance matrix is real symmetric and therefore always diagonalisable, with an orthonormal eigenvector basis. Diagonalisation ranks directions by variance, so the leading eigenvectors provide an optimal low-rank representation of measurement data and the trailing ones identify redundancy in an instrument array.

Computer graphics & geometry

Classifying and simplifying linear maps

A diagonalisable transformation is a pure scaling once the right axes are chosen. Recognising this lets a graphics or CAD pipeline replace a general matrix by three scale factors plus a basis change, which simplifies interpolation between transforms and reveals whether a mapping preserves, inverts or degenerates volume.

Electrical power systems

Symmetrical component transformation

Balanced three-phase network matrices are circulant, and the fixed transformation to positive, negative and zero sequence components diagonalises them. Diagonalisation converts a coupled three-phase fault calculation into three independent single-phase calculations, which is the standard basis of protection studies.

Design Considerations

Diagonalisability is not a numerical property

The predicate is discontinuous: an arbitrarily small perturbation can split a repeated eigenvalue and turn a defective matrix into a diagonalisable one. That diagonalisation will have an almost singular $S$ and is numerically useless. Decide on structure symbolically or from known matrix properties, never from a floating-point eigenvalue count.

Watch the conditioning of the eigenvector matrix

Errors in a diagonalisation are amplified by $\kappa(S) = \|S\| \, \|S^{-1}\|$ . Near-defective matrices have nearly parallel eigenvectors and a huge $\kappa(S)$ , so $A^{k} = SD^{k}S^{-1}$ can lose all significance even though every individual quantity is finite. Report $\kappa(S)$ alongside any computed diagonalisation.

Prefer the symmetric or Hermitian case where you can

A Hermitian matrix is always diagonalisable and admits a unitary $S$ , for which $\kappa(S) = 1$ exactly. If the model can be posed with a symmetric operator — by symmetrising a stiffness matrix, or by working with a covariance rather than a raw data matrix — both the existence question and the conditioning question disappear.

Do not form $S^{-1}$ when you only need to solve

Applying $A^{k}$ to a vector needs $S^{-1}\vec{v}$ , which is a linear solve rather than an inversion. Computing $S^{-1}$ explicitly costs more arithmetic and is less accurate. Factor $S$ once and reuse the factorisation across right-hand sides and across powers.

Choose the column order deliberately

Column order in $S$ is free, so use it. Ordering eigenvalues by decreasing modulus puts the dominant mode first, which is convenient for truncation and for reading off asymptotic behaviour. Grouping columns by eigenvalue keeps the diagonal blocks of $D$ contiguous, which matters when a downstream algorithm exploits block structure.

Have a fallback for defective matrices

When the multiplicity test fails, the options are the Jordan form (exact, but numerically fragile), the Schur decomposition $A = QTQ^{*}$ with $Q$ unitary and $T$ upper triangular (always exists, numerically stable, and enough for most computations), or the singular value decomposition when the question is about magnitudes rather than invariant directions. Production code should default to Schur.

Standards & Codes

Notation, interchange and numerical standards that govern how this material is written down, stored and computed in production systems.

Applicable standards, conventions and reference implementations
Reference	Title	Relevance to this topic
`ISO 80000-2`	Quantities and units — Part 2: Mathematics	Fixes the notation used throughout: italic for the matrix symbols $A$ , $S$ , $D$ , upright for the operators $\det$ and $\operatorname{diag}$ , and Greek $\lambda$ for eigenvalues.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Governs the rounding behaviour that makes an equality test between an algebraic and a geometric multiplicity meaningless in floating point, and that bounds the accuracy of any computed eigenvector matrix.
`LAPACK / BLAS reference`	Linear Algebra PACKage reference implementation	Provides `xGEEV` for the general eigenproblem and `xSYEV`/`xHEEV` for the symmetric and Hermitian cases; `xGEES` returns the Schur form, the standard fallback when a matrix is defective.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the matrix and multiplicity notation on this page semantically, so that a diagonal matrix is exposed to assistive technology as structured content rather than as an image.
`ISO 2041`	Mechanical vibration, shock and condition monitoring — Vocabulary	Defines the modal vocabulary — mode shape, natural frequency, modal coordinate — that maps directly onto the columns of $S$ and the diagonal entries of $D$ in structural applications.

Material Selection

For a mathematical topic, "material" is the numeric representation: the scalar field, storage format and precision the computation is built from.

Representation and precision selection
Representation	Select when	Trade-off
Exact rational arithmetic	Small matrices with integer or rational entries where diagonalisability itself is the deliverable, or where eigenvector bases feed a symbolic derivation.	Gives a definitive yes or no answer with no tolerance, but requires exact roots of the characteristic polynomial, which do not exist in closed form beyond degree four.
Algebraic number field arithmetic	Irrational eigenvalues must be handled exactly, for instance in exact modal analysis of a symmetric integer stiffness matrix.	Preserves exactness through the eigenvector computation, at a substantial implementation and runtime cost that grows quickly with the degree of the minimal polynomial.
IEEE 754 binary64 with a real spectrum	Symmetric or Hermitian matrices from measurement or discretisation, where a unitary diagonalisation is guaranteed to exist.	Backward stable and well conditioned because $\kappa(S) = 1$ , but eigenvalues that are close together still yield eigenvectors with large relative error even though the invariant subspace is accurate.
IEEE 754 binary64 with a complex spectrum	General non-symmetric system matrices in control and dynamics, where eigenvalues may be complex and eigenvectors are not orthogonal.	Requires complex storage and arithmetic for what may be a real matrix, and the accuracy of the result is limited by $\kappa(S)$ , which is unbounded near a defective matrix.
Real Schur form instead of a diagonalisation	Numerical work where existence must be guaranteed and stability matters more than a fully diagonal target.	Always exists with an orthogonal transforming matrix, so it is unconditionally stable, but leaves $2\times 2$ blocks for complex-conjugate pairs and an upper triangular remainder rather than a diagonal.
Finite field arithmetic	Coding theory and cryptanalysis, where a linear map over $\mathbb{Z}_p$ must be put in diagonal form.	Exact and fast, but the characteristic polynomial may not split over the field, so a matrix diagonalisable over $\mathbb{C}$ can fail to be diagonalisable over $\mathbb{Z}_p$ .

Manufacturing Notes

Implementation notes — how the result is actually produced by hand, by algorithm and by library, including cost and numerical behaviour.

Cost of the decision and the construction

Deciding diagonalisability by hand costs one characteristic polynomial plus one row reduction per distinct eigenvalue, so roughly $k$ reductions of an $n \times n$ matrix at about $\tfrac{2}{3}n^3$ operations each. Numerically the whole eigendecomposition is obtained in one pass: reduction to Hessenberg form followed by the shifted QR iteration costs about $10n^3$ operations for eigenvalues alone and around $25n^3$ when eigenvectors are also requested.

Verify with $AS = SD$ , not with $S^{-1}AS$

The multiplication-only identity $AS = SD$ needs no inverse, costs two matrix products, and isolates errors to a single column. Checking $S^{-1}AS$ instead introduces the inverse's own error and can mask a wrong eigenvector behind cancellation. In floating point, report the residual $\|AS - SD\|$ relative to $\|A\| \, \|S\|$ rather than testing for exact equality.

Library behaviour and conventions

numpy.linalg.eig returns eigenvalues and normalised right eigenvectors as the columns of a matrix that plays the role of

S

, but it never reports whether the matrix is defective — it simply returns a nearly singular eigenvector matrix. scipy.linalg.schur gives the always-available fallback. Symbolic systems expose an explicit predicate and a diagonalisation routine that raises an error for defective input, which is the behaviour to prefer when the structural answer matters.

Exploiting the diagonalisation for repeated application

If $A^{k}\vec{v}$ is needed for many values of $k$ , factor once: compute $\vec{z} = S^{-1}\vec{v}$ by a linear solve, then each power costs $n$ scalar multiplications plus one matrix-vector product $S(D^{k}\vec{z})$ , at $O(n^2)$ instead of $O(n^3)$ . When only the dominant behaviour is needed, truncating to the largest few $|\lambda_i|$ gives a reduced-order model with a controllable error bound.

Scaling and normalisation of eigenvectors

Eigenvectors are determined only up to a non-zero scalar, so $S$ is far from unique. Normalising each column to unit Euclidean length is the usual convention and improves the conditioning of the inverse; for a real matrix with a complex-conjugate eigenpair, fixing the phase (for instance by making the largest-magnitude component real and positive) makes results reproducible across libraries.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Assuming every square matrix is diagonalisable	high	Generalising from textbook examples with distinct eigenvalues, where the criterion is satisfied automatically.	Compute the eigenspace dimension for every repeated eigenvalue before asserting a diagonalisation exists; a repeated root is a question, not an answer.
Comparing only the algebraic multiplicities	high	Checking that the algebraic multiplicities sum to $n$ , which is always true and therefore tests nothing.	The criterion is $\gamma_A(\lambda) = \alpha_A(\lambda)$ for each eigenvalue separately; the geometric side requires an actual null space computation.
Mismatching the order of $S$ and $D$	high	Listing the eigenvalues in one order on the diagonal while the eigenvector columns were assembled in another.	Build $D$ only after $S$ is fixed, reading each diagonal entry off the eigenvalue of the corresponding column, then verify $AS = SD$ .
Using a dependent set of eigenvectors	medium	Taking several vectors from one eigenspace without checking that they form a basis, for instance including a scalar multiple twice.	Take eigenspace bases from a row reduction of $A - \lambda I_n$ , which produces independent vectors by construction, and confirm $\det(S) \ne 0$ .
Trusting a floating-point diagonalisation of a near-defective matrix	high	A computed eigenvector matrix is returned without error even when its condition number is $10^{12}$ , so $SD^{k}S^{-1}$ silently produces noise.	Always compute $\kappa(S)$ ; if it is large relative to the working precision, switch to a Schur decomposition or an invariant-subspace formulation.
Confusing similarity with row equivalence	medium	Attempting to diagonalise by row operations alone, which changes eigenvalues and destroys the spectrum.	A similarity transformation applies $S^{-1}$ on the left and $S$ on the right simultaneously; row reduction is used only inside the eigenspace computations, never on $A$ itself.
Assuming a real matrix has a real diagonalisation	medium	A real non-symmetric matrix may have complex-conjugate eigenvalues, so $S$ and $D$ are complex even though $A$ is not.	Expect complex arithmetic for general real matrices, or use the real Schur form with $2\times 2$ blocks if real output is a hard requirement.
Reporting the diagonal entries as if the order were canonical	low	Two correct diagonalisations of the same matrix list the eigenvalues differently, and downstream code assumes a fixed position.	Sort the eigenvalues explicitly by a stated rule, such as decreasing modulus, and permute the columns of $S$ to match before publishing results.

FAQs

What is the difference between diagonalisable and invertible?

They are independent properties. The zero matrix is diagonal, hence diagonalisable, but not invertible; the shear $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ is invertible but not diagonalisable. Invertibility asks whether zero is an eigenvalue; diagonalisability asks whether the eigenvectors span the space.

Does a repeated eigenvalue prevent diagonalisation?

No. It only means the question must be answered by computation rather than by inspection. If the eigenspace of the repeated eigenvalue has dimension equal to its algebraic multiplicity, the matrix is diagonalisable; the identity matrix has one eigenvalue of multiplicity $n$ and is already diagonal.

Is the diagonalisation of a matrix unique?

No. The columns of $S$ can be reordered, which permutes the diagonal of $D$ , and each column can be rescaled by any non-zero scalar. Within an eigenspace of dimension greater than one, any basis at all may be used. What is unique is the multiset of diagonal entries, since those are the eigenvalues with their multiplicities.

How does diagonalisation relate to the singular value decomposition?

They answer different questions. Diagonalisation uses a similarity transformation $S^{-1}AS$ , preserves eigenvalues, applies only to square matrices and may not exist. The SVD uses two different unitary matrices, $U^{*}AV$ , applies to any rectangular matrix, always exists and reveals magnitudes rather than invariant directions. They coincide, up to signs, for a symmetric positive definite matrix.

What happens when a matrix cannot be diagonalised?

The closest attainable form under similarity is the Jordan form: block diagonal with each block having one eigenvalue on the diagonal and ones on the superdiagonal. It is exact but extremely sensitive to perturbation, so numerical work uses the Schur decomposition $A = QTQ^{*}$ with $Q$ unitary and $T$ upper triangular instead, which always exists and is computed stably.

Why is

A^{k} = SD^{k}S^{-1}

so much cheaper than repeated multiplication?

Because the diagonal of $D^{k}$ is just $\lambda_i^{k}$ , so the entire power is $n$ scalar exponentiations plus two matrix products, independent of $k$ . Repeated squaring of $A$ costs $O(n^3 \log k)$ operations, and naive repeated multiplication $O(n^3 k)$ . The saving grows with $k$ , and the same substitution extends to $e^{At}$ and to any analytic function of the matrix.

Can I tell a matrix is diagonalisable without computing eigenvectors?

Sometimes. A Hermitian or real symmetric matrix always is; so is any matrix with $n$ distinct eigenvalues, which can be confirmed from the characteristic polynomial alone; and more generally any normal matrix, satisfying $A^{*}A = AA^{*}$ , is unitarily diagonalisable. Outside these classes, a repeated eigenvalue forces an eigenspace dimension computation.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section SD, subsection D. Licensed under the GNU Free Documentation License v1.2.
Golub, G. H. and Van Loan, C. F. Matrix Computations, 4th edition. Johns Hopkins University Press, 2013. Chapters 7 and 8 on the unsymmetric and symmetric eigenproblems.
Horn, R. A. and Johnson, C. R. Matrix Analysis, 2nd edition. Cambridge University Press, 2013.
Anderson, E. et al. LAPACK Users' Guide, 3rd edition. Society for Industrial and Applied Mathematics, 1999.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
Moler, C. and Van Loan, C. Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review 45(1), 2003.

AI Suggested Questions

Give me a 4x4 matrix that is not diagonalisable, show why the multiplicity test fails, and write out its Jordan form.
How large can the condition number of the eigenvector matrix become as a matrix approaches a defective one, and what does that do to a computed $A^{k}$ ?
Compare diagonalisation, Schur decomposition and singular value decomposition on the same non-symmetric matrix and explain what each one reveals.
Show how diagonalisation solves the coupled system $\frac{d\vec{x}}{dt} = A\vec{x}$ in closed form and what changes when $A$ is defective.
Two matrices have the same characteristic polynomial but only one is diagonalisable. Construct such a pair and explain what distinguishes them.
Work through the symmetrical component transformation for a balanced three-phase impedance matrix and identify the eigenvalues it produces.

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