PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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The non-negative square root of a non-negative operator is a polynomial in the operator

Statement

Let $T$ be a non-negative operator on a finite-dimensional real or complex inner product space. Then its unique non-negative square root is $p (T)$ for some polynomial $p \in R [x] \subseteq C [x]$ .

Facts & Assumptions

Given: A finite-dimensional real or complex inner product space $V$ , a non-negative endomorphism $T : V \to V$ , and the distinct eigenvalues $λ_{1}, \dots, λ_{r}$ of $T$ .

[L1]

A non-negative operator has an orthonormal eigenbasis with nonnegative eigenvalues (A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S).

[L2]

A non-negative operator has a unique non-negative square root (A non-negative operator has a unique non-negative square root).

Proof

technique · direct

1.1

By [L1], $V$ has an orthonormal eigenbasis consisting of eigenvectors of $T$ , and every eigenvalue $λ_{j}$ is a nonnegative real number. Interpolation on the finite set ${λ_{1}, \dots, λ_{r}}$ gives a polynomial $p \in R [x]$ with $p (λ_{j}) = \sqrt{λ_{j}}$ for every $j$ .

L1algebra

2.1

On each eigenspace $E_{λ_{j}} (T)$ , the operator $p (T)$ acts as multiplication by $p (λ_{j}) = \sqrt{λ_{j}}$ . Therefore $p (T)^{2}$ acts as multiplication by $λ_{j}$ , so $p (T)^{2} = T$ ; the same eigenbasis shows that $p (T)$ is non-negative. By [L2], $p (T)$ is the unique non-negative square root of $T$ .

L1L2step 1.1∎

Depends on

Used by

The non-negative square root of an explicit matrix is exhibited as a polynomial in the matrix Example

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, fourth edition (standard reference, not scraped)