A Mirror Reverses One Axis, and It Is Not Left or Right

Raise your right hand at a mirror and the reflected hand stays on the same side of the room, which means the usual question has a false premise. A plane mirror is the matrix diag(1, 1, -1): it fixes both axes lying in the glass and reverses only the direction you look along. Its determinant is -1, so no rotation reproduces it, and the sideways flip everyone reports belongs to the half turn you perform in your head.

Lambdia
August 15, 2026 · 11 min read

Raise your right hand in front of a mirror. The hand that comes up in the glass is on the same side of the room as yours. Nothing moved sideways. Yet everybody says a mirror swaps left and right, and the follow up question, why it does not also swap up and down, has kept the argument alive for a very long time.

The question has no answer because the premise is false. A mirror reverses one direction out of three, and it is neither of the two you were asked about. It reverses the direction you are looking along.

One matrix, three axes

Set up coordinates so the claim becomes checkable instead of arguable. Take $x$ along the room’s left to right axis, $y$ pointing up, and $z$ the direction you face, with the glass sitting in the plane $z = 0$ . A mirror in that position sends $(x, y, z)$ to $(x, y, -z)$ , which is a diagonal matrix:

M = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}

(1)

Apply it to the three basis vectors and the accounting is finished in a line. The vector pointing to your left comes back unchanged. The vector pointing up comes back unchanged. The vector pointing into the glass comes back negated.

M e_1 = e_1, \qquad M e_2 = e_2, \qquad M e_3 = -e_3

(2)

One axis reversed, two fixed, and the two that survive are the two lying in the glass. There is nothing about the situation that protects left and right in particular. The glass has a preferred direction and it is the only preferred direction in the room. Left and right are not special to the mirror. They are special to you.

Fig. 1 — Try all three directions. Two of them survive the glass untouched, and the one that does not is the one you look along.

Why you cannot turn into your reflection

Somebody will now say that your reflection is simply you, turned around. Turning around is a rotation, every rotation of space preserves orientation, and orientation preserving means determinant $+1$ . The mirror does not have that determinant.

\det M = -1 \;\neq\; +1 = \det R \quad \text{for every rotation } R

(3)

Determinants multiply, so nothing built out of rotations will ever produce $M$ . No sequence of turns performed by you, by the mirror or by anybody else superposes you on your reflection. That is the precise content of the observation that a right hand reflects to a left hand as an object rather than as a point of view, and it is why a right glove will not fit the reflected hand.

The trace is not sharp enough to see this, which is worth knowing before you reach for it. A rotation through $\theta$ has trace $1 + 2\cos\theta$ , so a quarter turn has trace $1$ , and so does $M$ . Only the determinant separates the two families, and it separates them completely: it takes the value $+1$ on the rotations and $-1$ on every other element of the orthogonal group.

One more fact, cheap to state and often useful: $M^2 = I$ . A reflection undoes itself, which is why the reflection of your reflection is you again.

Where the sideways feeling comes from

None of this makes the sideways feeling imaginary. It is real, and it has a location. To compare yourself with your reflection you have to face it, and facing it means performing a half turn about the vertical axis inside your head. That half turn is a genuine rotation with determinant $+1$ , so it smuggles no second reflection into the argument. Compose it with the mirror and look at the residue.

\operatorname{diag}(-1,1,-1)\,\operatorname{diag}(1,1,-1) = \operatorname{diag}(-1,1,1)

(4)

A pure left to right flip, with nothing else attached. The sideways reversal belongs to your half turn and not to the glass. If that really is the mechanism, then rotating some other way ought to leave a different residue behind.

\operatorname{diag}(1,-1,-1)\,\operatorname{diag}(1,1,-1) = \operatorname{diag}(1,-1,1)

(5)

Turn about the left to right axis instead, which is what you do by lying on your side, and the same mirror appears to flip up and down. Same glass, different leftover. The experiment costs nothing and it is the quickest way to convince somebody who will not look at a matrix.

Fig. 2 — The glass contributes the same column in both rows. The flip you notice is whichever one your own rotation failed to cancel.

The other decisive experiment needs a transparent sheet. Write a word on acetate and hold it up facing the mirror rather than turning it round. It stays readable, because reading uses only $x$ and $y$ and the mirror touched neither of them. What reverses writing on paper is you, flipping the paper over.

Any mirror, in any position

Reflection in a plane

For a plane through the origin with unit normal $n$ , the reflection is $H = I - 2nn^{\mathsf T}$ . It fixes every vector lying in the plane and sends $n$ to $-n$ , so its eigenvalues are $1, 1, -1$ and its determinant is $-1$ whatever the plane happens to be.

The diagonal matrix in equation (1) looked special only because the axes were chosen to flatter it. Take a normal pointing somewhere awkward, say $n = \tfrac{1}{3}(2, -1, 2)$ , and the reflection has no zero entry anywhere:

H = I - 2nn^{\mathsf T} = \frac{1}{9}\begin{pmatrix} 1 & 4 & -8 \\ 4 & 7 & 4 \\ -8 & 4 & 1 \end{pmatrix}, \qquad \operatorname{tr} H = 1, \quad \det H = -1

(6)

It still fixes a two dimensional plane and still negates exactly one direction. The count of reversed axes belongs to the plane rather than to the coordinates, so no relabelling of the room can turn one reversal into two. Tilt your bathroom mirror and the axis it reverses tilts with it, while two of your three directions carry on surviving and the handedness carries on changing.

Two mirrors, three mirrors, and the flat case

The edges of the argument are more interesting than the puzzle. Stand two mirrors at right angles and the composition of the two reflections has determinant $(-1)(-1) = +1$ , so it is a rotation. The image in a corner reflector really is you turned around, and you could in principle walk into that position. Add a third perpendicular mirror and the product is $-I$ , the point inversion, back to determinant $-1$ and back to swapped handedness.

\underbrace{\operatorname{diag}(1,1,-1)\operatorname{diag}(1,-1,1)}_{\det = +1} = \operatorname{diag}(1,-1,-1), \qquad \underbrace{\operatorname{diag}(1,1,-1)\operatorname{diag}(1,-1,1)\operatorname{diag}(-1,1,1)}_{\det = -1} = -I

(7)

Parity is the whole story. An even number of mirrors gives you a rotation, an odd number gives something no rotation can reach, and the statement about a single mirror is the case $k = 1$ of that.

Two places where the reasoning genuinely stops. Dimension matters, because in the plane $-I$ is a half turn with determinant $+1$ , so the sentence about point inversion is simply false there and the two dimensional analogue has to be argued again from scratch. And a curved mirror is not a linear map at all, so the eigenvalues have nothing to say about it. Whatever the back of a spoon is doing to your face, it is not this.

Sources and further reading

The phenomenon itself, and the standard confusions around it — Mirror image
The matrix $I - 2nn^{\mathsf T}$ and its uses in numerical linear algebra — Householder transformation
Where the split into determinant $+1$ and $-1$ lives — Orthogonal group
Transformations that are not rotations, in the language chemists use — Improper rotation
Handedness as a mathematical property rather than a convention — Chirality (mathematics)

Every matrix above was checked twice, once symbolically and once by walking the 48 signed permutation matrices that make up the symmetries of a cube. Exactly 24 of them are rotations, and the mirror is not one of the 24.

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