Trace, Determinant, and Matrix Functions

How trace and determinant summarize spectrum, and how functions like square root, inverse, logarithm, and exponential act on matrices through their eigenvalues.

Keywords

trace, determinant, matrix exponential, matrix logarithm, matrix square root

1 Role

This is the fifth page of the Matrix Analysis module.

The earlier pages built the operator, PSD, spectral, and perturbation language.

This page closes the first pass by showing how spectral information gets compressed into scalar summaries and functional calculus.

The key shift is:

once eigenvalues matter, many scalar functions can be applied to matrices through the spectrum

2 First-Pass Promise

Read this page after Perturbation and Stability.

If you stop here, you should still understand:

• why trace and determinant are spectral summaries
• how symmetric matrix functions are defined by acting on eigenvalues
• when square root, inverse, logarithm, and exponential make sense
• why these objects appear in ODEs, Gaussian models, kernels, and optimization

3 Why It Matters

Many advanced formulas are really about trace, determinant, or a matrix function:

• trace summarizes total spectral mass
• determinant summarizes signed volume scaling and invertibility
• log det appears in Gaussian likelihoods and barrier methods
• \(A^{-1}\) and \(A^{-1/2}\) appear in whitening and preconditioning
• \(e^{tA}\) appears in linear dynamics and diffusion

Without the spectral viewpoint, these can feel like disconnected tricks.

With the spectral viewpoint, they are all versions of the same idea:

understand the matrix by understanding what happens to its eigenvalues

4 Prerequisite Recall

• symmetric matrices admit orthonormal eigendecompositions
• PSD and positive definite matrices have nonnegative or positive eigenvalues
• perturbation theory tells us when spectral summaries are stable
• eigenvalues and singular values are operator-level information, not entrywise information

5 Intuition

5.1 Trace And Determinant As Spectral Summaries

The trace adds eigenvalues.

The determinant multiplies eigenvalues.

So they are not arbitrary formulas: they compress the spectrum in two different ways.

5.2 Matrix Functions From Scalar Functions

If a symmetric matrix diagonalizes as

\[ A = Q \Lambda Q^\top, \]

then any reasonable scalar function \(f\) can be applied to the diagonal entries of \(\Lambda\):

\[ f(A)=Q f(\Lambda) Q^\top. \]

This is the cleanest first-pass definition of a matrix function.

5.3 Domain Restrictions Matter

Not every scalar function makes sense on every spectrum.

For example:

• \(A^{-1}\) needs nonzero eigenvalues
• \(\log(A)\) needs positive eigenvalues
• \(A^{1/2}\) needs nonnegative eigenvalues if we want a real symmetric square root

So the eigenvalue domain tells you when the matrix function is legal.

6 Formal Core

Theorem 1 (Theorem Idea: Trace Is The Sum Of Eigenvalues) For a square matrix, the trace equals the sum of its eigenvalues, counted with algebraic multiplicity:

\[ \operatorname{tr}(A)=\sum_i \lambda_i. \]

In the symmetric case, this is especially transparent because the eigendecomposition is orthogonal.

Theorem 2 (Theorem Idea: Determinant Is The Product Of Eigenvalues) For a square matrix,

\[ \det(A)=\prod_i \lambda_i. \]

In particular, \(A\) is invertible if and only if \(\det(A)\neq 0\).

This is the spectral version of the volume-scaling picture.

Theorem 3 (Theorem Idea: Trace Is Cyclic) Whenever the products are defined,

\[ \operatorname{tr}(AB)=\operatorname{tr}(BA). \]

At first pass, this is one of the most useful algebraic identities involving trace.

Definition 1 (Definition: Matrix Function For A Symmetric Matrix) If

\[ A=Q\Lambda Q^\top \]

with \(Q\) orthogonal and \(\Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_n)\), then for a scalar function \(f\) defined on the eigenvalues,

\[ f(A)=Q\,\operatorname{diag}(f(\lambda_1),\dots,f(\lambda_n))\,Q^\top. \]

This is the cleanest first-pass route to matrix square roots, inverses, logarithms, and exponentials.

Theorem 4 (Theorem Idea: SPD Matrices Support The Most Useful First-Pass Functions) If \(A\) is symmetric positive definite, then the following are well-defined through the spectrum:

• \(A^{-1}\)
• \(A^{1/2}\)
• \(\log(A)\)
• \(\exp(A)\)

and they preserve the same eigenvectors while transforming the eigenvalues entrywise.

For example, if \(A=Q\Lambda Q^\top\) with \(\lambda_i>0\), then

\[ \log(A)=Q\,\operatorname{diag}(\log \lambda_i)\,Q^\top. \]

7 Worked Example

Let

\[ A= \begin{bmatrix} 4 & 0\\ 0 & 1 \end{bmatrix}. \]

Then the eigenvalues are \(4\) and \(1\), so

\[ \operatorname{tr}(A)=4+1=5, \qquad \det(A)=4\cdot 1=4. \]

Because \(A\) is symmetric positive definite, several matrix functions are immediately available:

\[ A^{1/2}= \begin{bmatrix} 2 & 0\\ 0 & 1 \end{bmatrix}, \qquad A^{-1}= \begin{bmatrix} 1/4 & 0\\ 0 & 1 \end{bmatrix}, \]

\[ \log(A)= \begin{bmatrix} \log 4 & 0\\ 0 & 0 \end{bmatrix}, \qquad \exp(A)= \begin{bmatrix} e^4 & 0\\ 0 & e \end{bmatrix}. \]

So one eigendecomposition gives many useful objects at once.

This is the matrix-function viewpoint in its simplest form.

8 Computation Lens

When a theorem contains trace, determinant, or a matrix function, ask:

1. is this really a statement about the eigenvalues?
2. is the matrix symmetric or PSD/PD?
3. is the function defined on the whole spectrum?
4. do we want a scalar summary like trace or log det, or a transformed matrix like \(A^{-1/2}\) or \(e^{tA}\)?

Those questions usually expose the underlying linear-algebra structure.

9 Application Lens

9.1 Linear Dynamics

The matrix exponential \(e^{tA}\) solves linear systems of ODEs and turns eigenvalue structure into growth and decay rates.

9.2 Gaussian Models And Optimization

Covariance matrices are SPD, so trace, inverse, and log det appear naturally in Gaussian likelihoods, regularization, and barrier methods.

9.3 Kernels, Whitening, And Geometry

Matrix square roots and inverse square roots appear in whitening, Mahalanobis geometry, kernel normalization, and covariance transport.

10 Stop Here For First Pass

If you can now explain:

• why trace sums eigenvalues
• why determinant multiplies eigenvalues
• how a symmetric matrix function is defined spectrally
• why SPD structure is what makes inverse, square root, and logarithm especially clean

then this page has done its job.

11 Go Deeper

This page closes the first-pass Matrix Analysis spine.

The strongest adjacent live pages right now are:

• Perturbation and Stability
• Optimization
• Kernel Ridge and Gaussian-Process Intuition
• Random Matrices and Spectral Concentration

12 Optional Deeper Reading After First Pass

The strongest current references connected to this page are:

• MIT 18.06 lecture notes - official notes for trace, determinant, eigenvalues, and diagonalization background. Checked 2026-04-25.
• MIT 18.03SC Matrix Exponentials - official unit page connecting matrix functions to linear dynamics. Checked 2026-04-25.
• The Exponential Matrix notes - official notes for the cleanest first-pass route to the matrix exponential. Checked 2026-04-25.
• Stanford EE364a: Convex Optimization I - official current course page where trace and log det repeatedly appear in convex and semidefinite modeling. Checked 2026-04-25.

13 Sources and Further Reading

• MIT 18.06 lecture notes - First pass - official notes for trace, determinant, eigenvalue, and diagonalization background. Checked 2026-04-25.
• MIT 18.03SC Matrix Exponentials - First pass - official unit page connecting spectral structure to matrix exponentials. Checked 2026-04-25.
• The Exponential Matrix notes - Second pass - official notes for the matrix-exponential viewpoint used in ODEs and linear systems. Checked 2026-04-25.
• Stanford EE364a: Convex Optimization I - Second pass - official course page where trace and log det feed into optimization models and PSD reasoning. Checked 2026-04-25.