Similar Matrices

Q: Why is the matrix $S$ required to be non-singular?

Because the definition uses $S^{-1}$, which exists only for a non-singular matrix. Structurally, non-singularity is what makes the columns of $S$ a basis, and hence what makes the transformation a genuine change of coordinates rather than a projection that loses information. A singular $S$ would collapse directions and could not be undone.

Q: Does it matter whether I say $A$ is similar to $B$ or $B$ is similar to $A$?

No. If $A = S^{-1}BS$ then $B = SAS^{-1}$, which is the same relation with $S^{-1}$ in place of $S$, and $S^{-1}$ is non-singular. The relation is symmetric, so the two statements are interchangeable and it is normal to say simply that the matrices are similar.

Overview

Row equivalence relates two matrices that describe the same system of equations. Similarity does something analogous for square matrices treated as operators: it relates two matrices that describe the same linear map, seen through two different bases. The definition is compact — $A = S^{-1}BS$ for some non-singular $S$ — but every occurrence of a change of coordinates in linear algebra, from modal analysis to the Jordan form, is an instance of it.

The requirement that $S$ be non-singular carries the whole meaning. A non-singular matrix has independent columns, so its columns form a basis of $\mathbb{C}^n$ , and multiplying by $S^{-1}$ translates a vector's ordinary coordinates into coordinates relative to that basis. The composite $S^{-1}BS$ therefore reads right to left as: convert from the new coordinates back to the standard ones, apply $B$ , then convert the result into the new coordinates. The net effect is the action of $B$ expressed entirely in the new frame.

Because similarity is a re-description rather than a change of behaviour, any quantity that is intrinsic to the behaviour must survive it. Eigenvalues do, along with the whole characteristic polynomial, the trace, the determinant, the rank and the multiplicity structure. Quantities that depend on the chosen coordinates — individual entries, symmetry, orthogonality of columns, banded structure — do not, and this is exactly what makes similarity useful: the entries can be simplified while the intrinsic content is held fixed.

The programme that follows from the definition is to choose $S$ so that $S^{-1}BS$ is as simple as possible. When $B$ has a full set of independent eigenvectors, taking them as the columns of $S$ produces a diagonal matrix, and every subsequent computation — determinant, powers, characteristic polynomial, exponential — becomes arithmetic on scalars. When it does not, the best available simple form retains a small amount of off-diagonal structure. Either way, the search for a good $S$ is the practical content of the subject.

Definition

SIM

Let $A$ and $B$ be square matrices of the same size $n$ . Then $A$ and $B$ are similar when there exists a non-singular matrix $S$ of size $n$ with $A = S^{-1}BS$ . When the matrix $S$ is to be emphasised, the phrasing is $A$ is similar to $B$ via $S$ .

The relation is symmetric: if $A = S^{-1}BS$ then $B = SAS^{-1} = (S^{-1})^{-1}A(S^{-1})$ , so $B$ is similar to $A$ via $S^{-1}$ . It is therefore safe to say simply that two matrices are similar, without ordering them.

Similarity Transformation

The operation $B \mapsto S^{-1}BS$ carried out by a fixed non-singular matrix $S$ is a similarity transformation. Applying it produces a matrix similar to $B$ ; applying it with $S^{-1}$ in place of $S$ reverses it exactly.

Also called conjugation by $S$ . The order matters: $S^{-1}BS$ and $SBS^{-1}$ are both similar to $B$ but are generally different matrices from each other.

Diagonalisable Matrix

DZM

A square matrix is diagonalisable when it is similar to a diagonal matrix, that is when a non-singular $S$ exists for which $S^{-1}BS$ has zero in every off-diagonal position.

Not every matrix is diagonalisable. The obstruction is a shortage of independent eigenvectors, which is exactly the condition that some eigenvalue has geometric multiplicity below its algebraic multiplicity.

Concepts

Similarity is a change of basis

Let $S$ be non-singular with columns $\vec{s}_1, \ldots, \vec{s}_n$ . Because $S$ is non-singular those columns are linearly independent and therefore a basis of $\mathbb{C}^n$ . For any vector $\vec{y}$ , the product $S\vec{y}$ is the linear combination of the columns with coefficients taken from $\vec{y}$ , so $S$ maps coordinates-in-the-new-basis to ordinary coordinates and $S^{-1}$ maps back. Reading $S^{-1}BS$ from right to left therefore gives: interpret the input as new-basis coordinates, apply $B$ in the standard frame, and re-express the output in the new basis. The matrix $A = S^{-1}BS$ is the same operator, written in a different language.

Eigenvectors transform with the basis

Suppose $A = S^{-1}BS$ and $B\vec{x} = \lambda\vec{x}$ . Then $A\left(S^{-1}\vec{x}\right) = S^{-1}BS S^{-1}\vec{x} = S^{-1}B\vec{x} = \lambda\left(S^{-1}\vec{x}\right)$ . So $S^{-1}\vec{x}$ is an eigenvector of $A$ for the same eigenvalue $\lambda$ . Eigenvalues survive a similarity transformation untouched; eigenvectors survive in a transformed form, carried across by $S^{-1}$ . Since $S^{-1}$ is non-singular it maps independent sets to independent sets, so the dimension of each eigenspace is preserved as well — similarity cannot create or destroy a defect.

Choosing $S$ to reach a diagonal form

Suppose $B$ has $n$ linearly independent eigenvectors $\vec{x}_1, \ldots, \vec{x}_n$ with eigenvalues $\lambda_1, \ldots, \lambda_n$ , and let $S$ be the matrix with those eigenvectors as its columns. Independence makes $S$ non-singular. Then $B S$ has columns $B\vec{x}_i = \lambda_i \vec{x}_i$ , and multiplying on the left by $S^{-1}$ sends each $\vec{x}_i$ to the standard basis vector $\vec{e}_i$ , giving columns $\lambda_i \vec{e}_i$ . The result is the diagonal matrix of eigenvalues. The construction is entirely explicit and it is the reason eigenvector computation matters.

What a simple similar form buys

If $A = S^{-1}BS$ is diagonal or triangular, the determinant is the product of the diagonal entries, the characteristic polynomial is already factored, the eigenvalues are visible without any computation, and the eigenvectors of $A$ are the standard basis vectors. Powers are trivial: $A^{k} = S^{-1}B^{k}S$ , so $B^{k} = SA^{k}S^{-1}$ and a diagonal $A$ makes $A^{k}$ a matter of raising $n$ scalars to the power $k$ . This transfer of an expensive matrix computation into cheap scalar arithmetic is the entire economic case for finding a good $S$ .

Similarity against row equivalence

Both relations partition matrices into classes, but they answer different questions and must not be conflated. Row equivalence applies to matrices of any shape and permits arbitrary independent row operations, corresponding to left multiplication by a non-singular matrix alone: $B \mapsto JB$ . Similarity applies only to square matrices and applies the inverse operation simultaneously on the right: $B \mapsto S^{-1}BS$ . Row equivalence preserves the row space and the null space but destroys the eigenvalues; similarity preserves the eigenvalues but destroys row-echelon structure. Every square matrix is row-equivalent to a diagonal matrix; not every square matrix is similar to one.

What similarity does not preserve

The individual entries carry no invariant meaning, and neither do several structural properties that engineers often rely on. Symmetry is not preserved — a symmetric $B$ conjugated by a general $S$ is usually not symmetric, which is why orthogonal or unitary $S$ is insisted upon whenever symmetry must survive. Sparsity, bandwidth and orthogonality of columns are all destroyed by a general similarity. The condition number is not preserved either, which is why an ill-conditioned $S$ can turn a benign problem into a numerically hazardous one.

Constructing and checking a similarity

Confirm both matrices are square and the same sizeSimilarity is undefined otherwise, since

S^{-1}BS

requires

S

and

B

to be conformable in both positions.

Choose or construct $S$ For a designed transformation, take the columns of

S

to be the new basis. For diagonalisation, take them to be

n

independent eigenvectors of

B

Verify $S$ is non-singularRow-reduce

S

to the identity, or confirm

\det(S) \ne 0

. A singular

S

has no inverse and no similarity is defined.

Compute $S^{-1}$ and form $S^{-1}BS$ Multiply right to left. Forming

B S

first and then applying

S^{-1}

keeps the intermediate quantities meaningful as transformed columns.

Check the invariantsTrace, determinant and characteristic polynomial must agree between

A

and

B

. Any disagreement means an arithmetic error, since these are theorems, not coincidences.

Equations

Definition of similarity

EQ-SM-01

A = S^{- 1} B S, S non-singular of size n

Both matrices must be square of size $n$ . The non-singularity of $S$ is what makes the relation reversible and meaningful.

Symmetry of the relation

EQ-SM-02

A = S^{- 1} B S \Leftrightarrow B = (S^{- 1})^{- 1} A (S^{- 1}) = S A S^{- 1}

Similarity in one direction gives similarity in the other, via the inverse matrix. The relation is therefore genuinely two-sided.

Transformation of eigenvectors

EQ-SM-03

B \vec{x} = λ \vec{x} \Rightarrow A (S^{- 1} \vec{x}) = λ (S^{- 1} \vec{x})

The eigenvalue is unchanged; the eigenvector is carried across by $S^{-1}$ . Independence and eigenspace dimensions are preserved because $S^{-1}$ is non-singular.

Diagonalisation by an eigenvector basis

EQ-SM-04

S = [{\vec{x}}_{1} ∣ {\vec{x}}_{2} ∣ \dots ∣ {\vec{x}}_{n}], S^{- 1} B S = [\begin{matrix} λ_{1} & 0 & \dots & 0 \\ 0 & λ_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & λ_{n} \end{matrix}]

Available exactly when $B$ has $n$ linearly independent eigenvectors. The eigenvalues appear in the same order as their eigenvectors in $S$ .

Powers under similarity

EQ-SM-05

A = S^{- 1} B S \Rightarrow A^{k} = S^{- 1} B^{k} S, B^{k} = S A^{k} S^{- 1}

The inner factors $SS^{-1}$ collapse between consecutive terms, so conjugation commutes with taking powers. This is the mechanism that makes diagonalisation valuable for repeated application.

Worked similarity transformation

EQ-SM-06

S = [\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}], S^{- 1} = [\begin{matrix} 0 & - 1 & 1 \\ - 1 & 0 & 1 \\ 1 & 1 & - 1 \end{matrix}]

A non-singular integer matrix with $\det(S) = -1$ , so its inverse is also integral and the transformation can be carried out exactly.

Preserved quantities

EQ-SM-07

tr (A) = tr (B), \det (A) = \det (B), rank (A) = rank (B)

Three of the invariants of similarity. Each provides an independent arithmetic check on a computed similarity transformation.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$A$	Transformed matrix	The result $S^{-1}BS$ of the similarity transformation	n x n complex matrix
$B$	Original matrix	The matrix being re-expressed in a new basis	n x n complex matrix
$S$	Similarity matrix	Non-singular matrix whose columns are the new basis	n x n non-singular
$S^{- 1}$	Inverse similarity matrix	Converts standard coordinates into coordinates relative to the columns of $S$	n x n non-singular
$n$	Size	Common order of all three matrices	positive integer
$λ$	Eigenvalue	Shared by similar matrices with equal multiplicities	complex number
$\vec{x}$	Eigenvector of $B$	Maps to the eigenvector $S^{-1}\vec{x}$ of $A$	non-zero vector in C^n
$tr (A)$	Trace	Sum of diagonal entries, invariant under similarity	complex number
$\det (S)$	Determinant of $S$	Must be non-zero for $S$ to be admissible	non-zero complex number

Worked Numerical Example

Problem statement

A symmetric stiffness matrix $B$ describes a three-degree-of-freedom model in physical coordinates. An analyst re-expresses the model in a non-orthogonal working basis given by the columns of $S$ . Compute the transformed matrix and verify that the two descriptions carry identical intrinsic content.

State the matrices
The physical matrix is symmetric and tridiagonal. The working basis is the set of columns of $S$ , chosen for convenience rather than orthogonality.
$B = [\begin{matrix} 7 & - 2 & 0 \\ - 2 & 6 & - 2 \\ 0 & - 2 & 5 \end{matrix}], S = [\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$
Confirm $S$ is non-singular and invert it
Expanding along the first row gives $\det(S) = 1(1 \cdot 1 - 1 \cdot 1) - 0 + 1(0 \cdot 1 - 1 \cdot 1) = -1$ , which is non-zero, so $S$ qualifies. Because the determinant is $\pm 1$ the inverse has integer entries and the whole computation stays exact.
$Confirm $S$ is non-singular and invert it$
Form $B S$ first
Multiplying on the right transforms the columns: column $j$ of $B S$ is $B$ applied to the $j$ -th new basis vector. Doing this step first keeps the intermediate result interpretable.
$B S = [\begin{matrix} 7 & - 2 & 5 \\ - 4 & 4 & 2 \\ 5 & 3 & 3 \end{matrix}]$
Apply $S^{-1}$ on the left
Left multiplication re-expresses each of those columns in the new basis. The result is the matrix of the same operator in the working coordinates.
$Apply $S^{-1}$ on the left$
Observe what changed
The transformed matrix is not symmetric, even though $B$ is. Symmetry is a property of the coordinate system, not of the operator, and only an orthogonal $S$ would have preserved it. The entries bear no visible resemblance to those of $B$ .
Check the invariants
The trace is $9 + 5 + 4 = 18$ for $A$ and $7 + 6 + 5 = 18$ for $B$ . Both determinants equal $162$ . Both characteristic polynomials are $-(x-3)(x-6)(x-9)$ , so the two matrices have the same eigenvalues with the same multiplicities.
$p_{A} (x) = p_{B} (x) = - (x - 3) (x - 6) (x - 9)$
Transport an eigenvector
The eigenvector of $B$ for $\lambda = 3$ is $(1,2,2)$ . Applying $S^{-1}$ gives $(0 - 2 + 2,\ -1 + 0 + 2,\ 1 + 2 - 2) = (0,1,1)$ , and direct multiplication confirms this is an eigenvector of $A$ for the same eigenvalue.
$A [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} - 1 + 1 \\ 5 - 2 \\ - 1 + 4 \end{matrix}] = [\begin{matrix} 0 \\ 3 \\ 3 \end{matrix}] = 3 [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}]$
Note the better choice of $S$
Had the columns of $S$ been the three eigenvectors $(1,2,2)$ , $(-2,-1,2)$ and $(2,-2,1)$ instead, the transformed matrix would have been $\operatorname{diag}(3,6,9)$ . The same operator, the same eigenvalues, but a representation in which every subsequent computation is scalar arithmetic. This is the target that the definition of similarity exists to make reachable.

Result

The two matrices $A$ and $B$ look nothing alike entry by entry, yet they share every intrinsic quantity: characteristic polynomial, eigenvalues, multiplicities, trace, determinant and rank. Only the coordinate-dependent features differ, and symmetry is one of them. The example also shows the cost of a poor choice of basis: a symmetric, sparse, well-conditioned matrix has become dense and non-symmetric with no analytical gain, which is why $S$ is normally chosen from eigenvectors or constrained to be orthogonal.

Applications & Industry Use

Structural engineering

Transformation to modal coordinates

Assembling the mode shapes of a structure as the columns of $S$ produces a similarity transformation that diagonalises the system matrix. In modal coordinates each degree of freedom evolves independently, so a many-degree-of-freedom transient becomes a set of uncoupled single-degree-of-freedom problems solved in parallel.

Control systems

State-space realisations

Any invertible change of state variables $\vec{z} = S^{-1}\vec{x}$ produces a new realisation with state matrix $S^{-1}AS$ . Controllable canonical, observable canonical and modal forms are all similarity transformations of one another, which is why controller designs based on pole locations transfer between realisations unchanged.

Computing & data

Numerical eigenvalue algorithms

The QR iteration is a sequence of orthogonal similarity transformations driving a matrix towards triangular form. Orthogonal similarity is chosen specifically because it preserves the two-norm and the condition number, so the eigenvalues of the final triangular matrix are those of the original to within a small backward error.

Mechanical engineering

Rotating a tensor into principal axes

Expressing a stress or inertia tensor in a rotated frame is a similarity transformation by the rotation matrix. Because rotation matrices are orthogonal, the transformation preserves symmetry as well as the eigenvalues, which is why principal stresses and principal moments of inertia are frame-independent quantities.

Chemical & process engineering

Reaction coordinate transformations

Recasting a linearised reaction network in terms of reaction extents rather than species concentrations is a change of basis on the state, and the Jacobian transforms by similarity. The relaxation time constants, being eigenvalues, are unchanged, confirming that the choice of coordinates does not alter the physics.

Robotics

Changing reference frames

A linear map expressed in a tool frame and the same map expressed in a world frame are related by conjugation with the frame transformation. Quantities used for planning — manipulability eigenvalues, condition numbers of the Jacobian in a normalised metric — must be checked for frame dependence, since only the similarity invariants transfer unchanged.

Design Considerations

Prefer an orthogonal or unitary $S$ wherever possible

When $S$ is orthogonal, $S^{-1} = S^{t}$ , so the inverse costs nothing, symmetry is preserved and the transformation neither amplifies nor attenuates error. A general $S$ has none of these properties. Every serious numerical eigenvalue algorithm restricts itself to orthogonal or unitary similarity for exactly this reason.

Watch the conditioning of $S$

The accuracy of a computed similarity transformation degrades with the condition number $\kappa(S)$ , and errors in $B$ are amplified by up to $\kappa(S)^2$ in the eigenvalues of $A$ . Near-defective matrices force an ill-conditioned eigenvector matrix, which is precisely the situation where the naive diagonalising $S$ should not be used.

Fix the direction convention and document it

Whether the transformation is written $S^{-1}BS$ or $SBS^{-1}$ determines whether the columns of $S$ are the new basis or the old one. Both conventions appear in the literature and in software. A single unstated convention change inverts the transformation and produces plausible but wrong results, so state it explicitly in interfaces and documentation.

Do not expect structure to survive

Symmetry, bandedness, sparsity, non-negativity and stochasticity are all coordinate-dependent and are generally destroyed by a similarity transformation. If a downstream algorithm relies on such a property, either constrain $S$ to preserve it — orthogonal for symmetry, permutation for sparsity patterns — or re-establish the property explicitly after the transformation.

Use similarity to move work, not to create it

The transformation is worth doing only when the transformed matrix makes some subsequent computation cheaper: repeated powers, a matrix exponential, many right-hand sides. For a single determinant or a single linear solve, the cost of forming $S^{-1}BS$ exceeds the saving. Decide what the transformed form is for before computing it.

Test similarity by invariants, but do not prove it by them

Equal trace, determinant and characteristic polynomial are necessary for similarity, so a mismatch definitively refutes it. They are not sufficient: two matrices can share a characteristic polynomial and still fail to be similar, when their eigenspace dimensions differ. Establishing similarity requires exhibiting an $S$ , or comparing canonical forms.

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	Sets the notation for the inverse superscript, the determinant and trace operators, and the bracket forms used to display the similarity transformation.
`LAPACK reference implementation`	Linear Algebra PACKage reduction and Schur routines	`xGEHRD` and `xHSEQR` implement the orthogonal similarity transformations that reduce a matrix to Hessenberg and then Schur form, the numerical backbone of every dense eigenvalue computation.
`IEC 61131-3`	Programmable controllers — Programming languages	State-space controller blocks implemented under this standard are realisation-dependent, so the similarity transformation between a design realisation and an implementation realisation must be recorded to interpret internal states.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Bounds the rounding error in forming $S^{-1}BS$ , which is amplified by the condition number of $S$ and is the reason non-orthogonal similarity is avoided numerically.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the matrix displays and inverse superscripts on this page as semantic markup, keeping the transformation searchable and accessible.

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
Unimodular integer $S$ with $\det(S) = \pm 1$	Exact hand or symbolic work where the inverse must also be integral, as in the worked example.	Keeps every intermediate quantity an integer with no rounding at all, but such matrices are rare and cannot generally be built from eigenvectors.
Exact rational $S$	Symbolic diagonalisation of a small matrix whose eigenvectors have rational entries.	Exact and fully general within its scope, but the entries of $S^{-1}$ can have large denominators and the transformed matrix becomes unwieldy.
Orthogonal or unitary $S$ in binary64	Any numerical similarity transformation, and mandatory when symmetry must be preserved.	Perfectly conditioned with $S^{-1} = S^{*}$ available for free, but cannot reach a diagonal form for a general non-normal matrix; the best attainable is triangular.
General eigenvector $S$ in binary64	The matrix is comfortably diagonalisable and a diagonal form is needed for powers or exponentials.	Gives the simplest possible transformed matrix, but the accuracy of everything downstream is governed by $\kappa(S)$ , which is unbounded as the matrix approaches defectiveness.
Permutation $S$	Reordering states or degrees of freedom, for example to expose block structure or improve sparsity.	Exact, cost-free to invert, and preserves sparsity and symmetry, but can only reorder — it cannot simplify the numerical content of the matrix.
Block-diagonal $S$	The model decomposes into weakly coupled subsystems that are each transformed independently.	Preserves the block structure and keeps the conditioning local to each block, but cannot remove coupling between blocks.

Manufacturing Notes

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

Cost of forming a similarity transformation

Computing $S^{-1}BS$ directly requires one inversion at roughly $2n^3$ operations and two matrix multiplications at $2n^3$ each, about $6n^3$ in total. Solving $SA = BS$ for $A$ by an LU factorisation of $S$ avoids the explicit inverse and costs about $\tfrac{2}{3}n^3 + 4n^3$ , which is both cheaper and more accurate. Never form an explicit inverse when a solve will do.

Order of multiplication matters for interpretation

Compute $B S$ first, then apply $S^{-1}$ . The intermediate $B S$ has columns equal to $B$ applied to each new basis vector, which is directly meaningful and easy to check by hand. Computing $S^{-1}B$ first gives an intermediate with no such reading, and errors are correspondingly harder to localise.

Verification of a computed transformation

Four cheap checks apply. Confirm $\operatorname{tr}(A) = \operatorname{tr}(B)$ and $\det(A) = \det(B)$ . Confirm $SA = BS$ by direct multiplication, which avoids relying on the computed inverse. And transport one known eigenvector of $B$ through $S^{-1}$ and verify it is an eigenvector of $A$ for the same eigenvalue.

Library behaviour

No mainstream library offers a single similarity-transformation call, because the useful operation is always the construction of a particular $S$ . numpy.linalg.eig returns the eigenvector matrix that diagonalises; scipy.linalg.schur returns an orthogonal or unitary $Z$ with $Z^{*}AZ$ triangular; scipy.linalg.hessenberg returns the orthogonal reduction used inside eigensolvers. Prefer these to hand-assembled transformations.

Numerical stability

An orthogonal similarity is backward stable: the computed result is the exact transformation of a matrix within roundoff of the input. A general similarity is not, and the error can be amplified by $\kappa(S)$ or worse. Where a non-orthogonal $S$ is unavoidable, report $\kappa(S)$ alongside the result so that downstream accuracy claims can be scaled accordingly.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Using a singular $S$	high	Assembling $S$ from eigenvectors of a defective matrix, so the columns are dependent.	Verify that $S$ row-reduces to the identity or that $\det(S) \ne 0$ before inverting. A defective matrix cannot supply $n$ independent eigenvectors.
Reversing the direction of the transformation	high	Computing $SBS^{-1}$ when $S^{-1}BS$ was intended, or misreading whether the columns of $S$ are the new or the old basis.	Fix one convention, write it into the interface documentation, and test it on a case with a known answer such as a permutation.
Expecting symmetry to survive	medium	Transforming a symmetric matrix by a general non-orthogonal $S$ and passing the result to a symmetric eigensolver.	Restrict $S$ to be orthogonal or unitary when symmetry must be preserved; otherwise dispatch to a general solver after the transformation.
Inferring similarity from equal eigenvalues	medium	Observing that two matrices share a characteristic polynomial and concluding they are similar.	Equal characteristic polynomials are necessary but not sufficient. Compare eigenspace dimensions, or exhibit an explicit $S$ .
Confusing similarity with row equivalence	medium	Applying row operations alone and expecting the eigenvalues to be preserved.	Row reduction is one-sided and destroys eigenvalues. Similarity requires the inverse operation applied simultaneously on the other side.
Forming an explicit inverse unnecessarily	low	Computing $S^{-1}$ as a matrix and multiplying, rather than solving $SA = BS$ .	Factor $S$ once and solve. The explicit inverse is more expensive and less accurate, and offers no advantage when it is used only once.
Ignoring an ill-conditioned $S$	high	Diagonalising a near-defective matrix with its eigenvector matrix and trusting the result.	Compute and report $\kappa(S)$ . Where it is large, use a Schur factorisation, which is always available and always well conditioned, in place of diagonalisation.
Applying similarity to non-square or mismatched matrices	low	Attempting $S^{-1}BS$ with $B$ rectangular or with $S$ of a different size.	Similarity is defined only for square matrices of equal size. For rectangular matrices the corresponding notion is equivalence, $B \mapsto PBQ$ with $P$ and $Q$ separately non-singular.

FAQs

Why is the matrix

S

required to be non-singular?

Because the definition uses $S^{-1}$ , which exists only for a non-singular matrix. Structurally, non-singularity is what makes the columns of $S$ a basis, and hence what makes the transformation a genuine change of coordinates rather than a projection that loses information. A singular $S$ would collapse directions and could not be undone.

Does it matter whether I say

A

is similar to

B

B

is similar to

A

No. If $A = S^{-1}BS$ then $B = SAS^{-1}$ , which is the same relation with $S^{-1}$ in place of $S$ , and $S^{-1}$ is non-singular. The relation is symmetric, so the two statements are interchangeable and it is normal to say simply that the matrices are similar.

Are similar matrices equal in any sense?

They are equal as descriptions of a linear operator, but not as arrays of numbers. Every entry can differ, and structural properties such as symmetry and sparsity can be lost. What is shared is everything intrinsic to the operator: eigenvalues, characteristic polynomial, trace, determinant, rank, and the dimensions of every eigenspace.

Is every square matrix similar to a diagonal matrix?

No. Diagonalisation requires $n$ linearly independent eigenvectors to fill the columns of $S$ , and a defective matrix — one with an eigenvalue whose geometric multiplicity is below its algebraic multiplicity — cannot supply them. The best general result is that every square matrix is similar to a Jordan form, which is diagonal apart from ones on the superdiagonal.

How is similarity different from row equivalence?

Row equivalence applies a non-singular matrix on the left only, $B \mapsto JB$ , and is defined for matrices of any shape. Similarity applies $S^{-1}$ on the left and $S$ on the right simultaneously, and requires square matrices. Row equivalence preserves the null space and row space but not the eigenvalues; similarity preserves the eigenvalues but not the echelon structure.

If two matrices have the same eigenvalues, must they be similar?

No. The identity matrix of size two and the matrix with ones on the diagonal and a single one above it share the characteristic polynomial $(x-1)^2$ , but the identity is similar only to itself, since $S^{-1}I_2S = I_2$ for every $S$ . Equal eigenvalues are necessary for similarity but not sufficient; the eigenspace dimensions must also agree.

Why do numerical algorithms insist on orthogonal similarity?

Because an orthogonal matrix satisfies $S^{-1} = S^{t}$ and has condition number exactly one. The transformation therefore costs no inversion, preserves the two-norm and symmetry, and does not amplify rounding error. A general similarity can amplify errors by the condition number of $S$ , which is unbounded, so it is used only when the resulting simplification justifies the risk.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section SD, Subsection SM. Licensed under the GNU Free Documentation License v1.2.
Horn, R. A. and Johnson, C. R. Matrix Analysis, 2nd edition. Cambridge University Press, 2012. Chapter 1 and Chapter 3.
Golub, G. H. and Van Loan, C. F. Matrix Computations, 4th edition. Johns Hopkins University Press, 2013. Section 7.1.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
Anderson, E. et al. LAPACK Users' Guide, 3rd edition. Society for Industrial and Applied Mathematics, 1999.

AI Suggested Questions

Show two matrices with identical characteristic polynomials that are not similar, and prove they are not by comparing eigenspace dimensions.
Why can a similarity transformation by an orthogonal matrix preserve symmetry when a general one cannot?
Derive the state matrix of a controllable canonical realisation as an explicit similarity transformation of a given state-space model.
How does the condition number of the eigenvector matrix bound the error in a numerically computed diagonalisation?
Explain why the identity matrix is similar only to itself, and characterise all matrices with that property.
What is the relationship between similarity of square matrices and equivalence of rectangular matrices under $B \mapsto PBQ$ ?

Related Calculators

Similarity Transformation Calculator

Compute $S^{-1}BS$ exactly or numerically and report the invariants shared by the two matrices.

Diagonalisability Checker

Decide whether a matrix admits a diagonalising $S$ and return the eigenvector matrix and its condition number.

Matrix Inverse Calculator

Invert the similarity matrix $S$ with a full audit trail, or solve $SA = BS$ without forming the inverse.

Overview

Definition