Rigid-Body Mechanics: Similarity Transformation of the Inertia Tensor

The inertia tensor is a physical property of a rigid body about a specified point, but the matrix used to represent that tensor depends on the coordinate frame. When the coordinate axes are rotated, the physical mass distribution has not changed; only its matrix representation has changed. The correct coordinate transformation is an orthogonal similarity transformation.

This article derives that result carefully and connects it to angular momentum, rotational kinetic energy, the mass-integral definition of inertia, and principal-axis diagonalization. The key result is

(1)

when C_BA maps vector components from frame A into frame B.

1 Video companion at the start of the derivation

Companion video for the inertia-tensor similarity-transformation derivation.

The video is placed here intentionally, immediately before the mathematical derivation, so that the PhysicsLibrary video-embedding macro can be tested in the same location where a learner would naturally use it.

2 What the inertia tensor represents

Consider a rigid body and a reference point O. Let r denote the position of a mass element dm relative to O. In an orthonormal Cartesian frame, the inertia tensor is represented by the symmetric matrix

(2)

where 1₃ is the 3 × 3 identity matrix [1, 2, 3].

Written component-by-component,

(3)

Depending on the convention used for products of inertia, some engineering texts write the off-diagonal entries as −I_xy, −I_xz, and −I_yz. The transformation law derived below is unchanged as long as the matrix convention is used consistently.

The tensor appears directly in three important physical relations. Angular momentum about O is

(4)

rotational kinetic energy is

(5)

and the scalar moment of inertia about a unit axis n through O is

(6)

These relations describe physical quantities and therefore cannot depend on an arbitrary choice of coordinate axes.

3 Coordinate-frame convention

Let frames A and B be orthonormal frames with the same origin O. Define the direction cosine matrix C_BA by

(7)

for every geometric vector v.

Because C_BA represents a proper orthogonal rotation,

(8)

and therefore

(9)

4 Derivation from angular momentum

This is the shortest physical derivation of the similarity transformation.

In frame A,

(10)

The geometric vectors H and ω transform according to the ordinary vector rule,

(11)

(12)

From Equation (9),

(13)

Substitute Equation (10) into Equation (11):

(14)

Now substitute Equation (13):

(15)

But in frame B the same physical angular-momentum law must be

(16)

Since Equations (15) and (16) must agree for arbitrary ω^B,

(17)

Finally, orthogonality gives

(18)

The matrix has to appear on both sides of I^A. A second-order tensor has two coordinate indices, so both indices must be transformed.

5 Why this is called a similarity transformation

In linear algebra, a similarity transformation has the form

(19)

Equation (17) has exactly this form with

(20)

For an orthogonal rotation,

(21)

so the transformation becomes

(22)

This expression is also called an orthogonal congruence transformation. For a rotation matrix, the distinction disappears algebraically because inverse and transpose are identical. Conceptually, however, the similarity viewpoint is especially useful because it immediately explains why the eigenvalues are preserved [4].

6 Independent derivation from the mass integral

The transformation law can also be derived directly from the definition of inertia.

Let

(23)

The Euclidean norm is unchanged by rotation:

(rB)T rB	= (rA)T C BAT C BArA	(24)
	= (rA)T rA.	(25)

The outer product transforms as

rB(rB)T

= C BArA(rA)T C BAT .

(26)

Insert these expressions into the inertia integral:

IB	= ∫ dm	(27)
	= ∫ dm.	(28)

Because

(29)

we may factor the rotation matrices outside the integral:

IB	= C BACBAT	(30)
	= CBAIAC BAT .	(31)

Thus the same transformation follows directly from the mass distribution itself.

7 Rotational kinetic energy is invariant

The same physical rigid body must have the same rotational kinetic energy no matter which frame is used to write the components.

Starting in frame B,

(32)

Substitute

(33)

and Equation (18):

T	= (ωA)T C BAT C BAIAC BAT C BAωA	(34)
	= (ωA)T IAωA.	(35)

Hence

(36)

The matrix entries may change, but the physical scalar does not.

8 Principal axes are the eigenvectors of the inertia tensor

Because the inertia tensor is real and symmetric, it possesses three mutually orthogonal eigenvectors and real eigenvalues. Let

(37)

where the columns of V are orthonormal eigenvectors expressed in frame A, and

(38)

contains the principal moments of inertia.

Since V is orthogonal,

(39)

Multiplying Equation (37) on the left by V^T gives

(40)

This is precisely the inertia-tensor similarity transformation. If frame P is chosen to have its axes along the principal axes, then

(41)

and

(42)

The products of inertia vanish in the principal-axis frame.

9 What the similarity transformation preserves

Similar matrices have the same characteristic polynomial. Therefore rotating the coordinate frame preserves

• the three eigenvalues, which are the principal moments of inertia;
• the trace;
• the determinant;
• the characteristic polynomial;
• positive-definite or positive-semidefinite character;
• rotational kinetic energy when the angular-velocity components are transformed consistently.

For example,

(43)

Using the cyclic property of the trace,

tr(IB)	= tr(IAC ABCBA)	(44)
	= tr(IA).	(45)

Likewise,

det(IB)	= det(C BA) det(IA) det(C AB)	(46)
	= det(IA).	(47)

These invariants are useful software checks after transforming an inertia matrix.

10 Worked numerical example: rotate a principal inertia tensor

Suppose the principal-axis inertia matrix is

(48)

Let frame A be rotated relative to the principal frame by 𝜃 = 30^∘ about the common z axis. Let

(49)

The inertia matrix expressed in frame A is

(50)

Writing c = cos 𝜃 and s = sin 𝜃,

(51)

At 𝜃 = 30^∘,

(52)

so

(53)

Nothing physical happened to the body. The off-diagonal entries appeared only because frame A is not aligned with the principal axes.

To recover the principal frame,

(54)

which returns Equation (48).

The invariants provide immediate checks:

(55)

while

(56)

The eigenvalues of I^A are likewise 2, 5, and 8 kg m².

11 Moment of inertia about a physical axis is also invariant

Let the physical axis be the x_A direction,

(57)

From Equation (53),

(58)

In the principal frame,

(59)

Then

In	= (nP )T IP nP	(60)
	= 2 cos 2𝜃 + 5 sin 2𝜃	(61)
	= 2.75 kg m2.	(62)

The components changed, but the physical moment of inertia about the chosen physical axis did not.

12 Similarity transformation versus the parallel-axis theorem

These two operations are often confused, but they solve different problems.

A similarity transformation changes the orientation of the coordinate basis while keeping the same reference point and the same physical inertia tensor:

(63)

The parallel-axis theorem changes the reference point about which the inertia is computed. If G is the center of mass and O is displaced from G by d, then

(64)

Equation (64) changes the tensor itself because the moment reference point has changed. Equation (18) changes only its coordinate representation.

If both the point and frame change, apply both operations, keeping the geometry and frame convention explicit.

13 Passive frame change versus active body rotation

There are two geometrically different situations that can produce the same matrix pattern.

In a passive frame change, the body remains fixed and the coordinate axes are changed. With the convention of Equation (7),

(65)

In an active rotation, the coordinate frame is held fixed while the physical body is rotated. If Q maps the old physical directions into the new physical directions in that fixed frame, then the newly oriented tensor is

(66)

The algebra looks the same, but the interpretation is different. Confusing active and passive meanings is a common source of transposes and sign errors.

14 Connection with covariance transformations

The same two-sided transformation appears in estimation theory. If a random vector transforms as

(67)

then its covariance transforms as

(68)

The reason is structurally identical: a covariance matrix, like the inertia tensor matrix, represents a second-order object with two coordinate indices. This analogy is especially useful in navigation, estimation, and rigid-body dynamics.

15 Common mistakes

• Transforming inertia like a vector, for example using only CI. A second-order tensor requires a two-sided transformation.
• Using C^T IC without first stating what direction C maps. Both orders occur in textbooks because DCM conventions differ.
• Treating the off-diagonal products of inertia as intrinsic properties that must have the same numerical values in every frame. They are coordinate-dependent components.
• Assuming diagonal inertia means a body is geometrically symmetric. Every real symmetric inertia tensor has orthogonal principal axes even for an asymmetric body.
• Confusing a coordinate rotation with the parallel-axis theorem.
• Forgetting to rotate vectors such as ω, H, or the axis vector n consistently when checking physical invariance.
• Calling every expression CIC^T a similarity transformation without noting that the inverse equals the transpose only for an orthogonal matrix.

16 Compact derivation chain

The essential logic can be summarized in four lines:

HA	= IAωA,	(69)
HB	= C BAHA,	(70)
ωA	= C ABωB,	(71)
∴ IB	= C BAIAC AB .	(72)

For orthonormal Cartesian frames,

(73)

Choosing B to be the eigenvector, or principal-axis, frame gives

(74)

Thus principal-axis diagonalization is not a separate trick. It is the similarity transformation applied with the special rotation that aligns the coordinate axes with the eigenvectors of the inertia tensor.

References