Can entries of sign(A) exceed one in magnitude?

Yes. The worked example contains an entry equal to four.

Does A have to be diagonalizable?

No. The matrix-function definition extends through Jordan form when the spectrum avoids the imaginary axis.

Are the resulting projectors always orthogonal?

No. They are spectral projectors and may be oblique.

AI Glossary / Mathematics and optimization / Intermediate

Matrix sign function

Definition

The matrix sign function maps a square matrix's spectral components in the open right half of the complex plane to +1 and those in the open left half to −1. In its standard half-plane definition, it requires that the matrix have no eigenvalues on the imaginary axis. It is a matrix function, not an instruction to replace each entry by its individual sign.

What it means

For a real scalar, positive and negative numbers lie on opposite sides of zero. For a complex eigenvalue, the analogous distinction uses the sign of its real part. The matrix function preserves the associated invariant subspaces while replacing the spectral values by +1 or −1.
The resulting matrix S satisfies S²=I, but its individual entries need not belong to {−1,0,1}. It may also be nonsymmetric even when all its eigenvalues are real. Its main value is structural: it can separate groups of modes and help construct projectors and algorithms for matrix equations.

How it works

For a diagonalizable matrix A=VΛV^−1, replace each diagonal eigenvalue of Λ by the sign of its real part, then transform back with V. A Jordan-form definition extends the function to matrices that are not diagonalizable, provided the spectrum avoids the imaginary axis.
If S=sign(A), then (I+S)/2 projects onto the right-half-plane invariant subspace, and (I−S)/2 onto the left-half-plane subspace. These projectors need not be orthogonal. Computational methods can obtain S without explicitly computing an eigenvector decomposition. One classical iteration starts with X_0=A and uses X_(k+1)=(X_k+X_k^−1)/2. Under the stated spectral condition it converges to the matrix sign in exact arithmetic, though numerical stability and scaling still matter.

Three practical examples

A control-system calculation can separate stable and unstable continuous-time modes according to whether their eigenvalues lie in the left or right half-plane.
A numerical algorithm can use a matrix sign computation as part of solving certain Sylvester, Lyapunov, or Riccati equations.
A scientific-computing developer can construct spectral projectors to isolate a selected invariant subspace for further analysis.

Think of it this way

Think of sorting a system's modes into two rooms separated by a vertical boundary. The sign function records which room each mode belongs to while retaining how those modes combine in the original coordinates. The analogy does not apply entry by entry: a matrix's displayed entries are not themselves its modes, and the boundary excludes eigenvalues exactly on the imaginary axis.

Worked example

Consider an original matrix A with rows (2,6) and (0,−1). Its eigenvalues are two and minus one. It has the decomposition A=VDV^−1 with V having rows (1,−2) and (0,1), and D=diag(2,−1). Replacing D by diag(1,−1) gives S with rows (1,4) and (0,−1).
Check S² directly: its upper-right entry is 1×4+4×(−1)=0, and both diagonal entries are one. Thus S²=I. By contrast, taking signs entry by entry gives rows (1,1) and (0,−1), which is not the correct matrix function. Merely squaring to I is insufficient to select the correct answer, since that incorrect matrix also squares to I.
The right-half-plane projector P_+=(I+S)/2 has rows (1,2) and (0,0). The left-half-plane projector P_−=(I−S)/2 has rows (0,−2) and (0,1). Applying them to v=(1,1)ᵀ gives (3,0)ᵀ and (−2,1)ᵀ. They add back to v. A maps the first result to twice itself and the second result to minus itself, confirming their respective invariant directions.
The iteration can also be checked. A^−1 has rows (0.5,3) and (0,−1), so X_1 has rows (1.25,4.5) and (0,−1). The next iterate has rows (1.025,4.05) and (0,−1), moving toward S. These exact small calculations illustrate convergence; they are not a timing comparison or a universal stopping rule.

Illustrative case study

In a fictional simulation team, an engineer needs to split a two-mode continuous-time model into growing and decaying components. A spreadsheet's ordinary elementwise sign operation initially looks plausible, but the projector check above reveals that it does not recover the required invariant directions. The team switches to a matrix-function routine and verifies the decomposition on a known example. This is an illustrative validation story, not a claim about a real engineering deployment.

Common misconceptions and mistakes

The scalar complex sign used here is determined by the real part, rather than z/|z|. An eigenvalue on the imaginary axis falls outside this standard definition; it should not silently be assigned a half-plane sign. Eigenvalues near the boundary can also make the subspace split sensitive to perturbations.
Another mistake is using the half-plane distinction as the stability test for a discrete-time system. Discrete-time asymptotic stability concerns the unit disk, so a different transformation or analysis is needed. Finally, the identities S²=I and SA=AS are useful checks but do not alone establish that the intended spectral signs were selected.

Business use

For business-facing engineering software, this function is usually an internal numerical tool. A useful review asks what spectral separation the application needs, whether the matrix satisfies the domain assumptions, and how sensitive the result is to data uncertainty. Report residuals and validation checks alongside any derived classification of modes. The calculation should support the model's intended analysis, not substitute for checking whether the model represents the process.

Check your understanding

What does P_+ do to the worked vector (1,1)ᵀ?

Answer: It produces (3,0)ᵀ, the component in the eigenvalue-two direction. The complementary projector supplies (−2,1)ᵀ, and the components sum to the original vector.

Related terms

Eigenvalue
Matrix function
Invariant subspace
Spectral projector
Jordan form
Newton iteration
Sylvester equation
Riccati equation

Learn next

Verify P_+²=P_+, P_−²=P_−, and P_+P_−=0 for the example. Then check why P_+ is not an orthogonal projector by comparing it with its transpose.

Sources

Sources: Numerical-analysis researcher Nicholas Higham's technical explanation: https://nhigham.com/2020/12/15/what-is-the-matrix-sign-function/ . Official SciPy matrix-sign documentation: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.signm.html . The example and projector calculations here are original.

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