The first structural class: operators that preserve length. It turns out preserving length forces preserving angle, forces $T^*T=I$, and forces the columns of the matrix to be an orthonormal basis. Four different-looking conditions, all the same condition.
These are the changes of coordinates that do not distort geometry. If you want to move to a nicer basis without changing lengths or angles — which is exactly what the spectral theorem does — the change-of-basis matrix must be unitary.
Rotation matrices in graphics and robotics are orthogonal. Quantum time evolution is unitary, which is what conserves total probability. The discrete Fourier transform is unitary after normalising, which is why it preserves energy — Parseval's theorem in signal processing.
The orthogonal group $O(n)$ and unitary group $U(n)$ are the first examples of Lie groups, the objects that organise modern physics. Their representation theory (MAT445H1) classifies the symmetries of quantum systems.
A linear $T:V\to V$ is an isometry if $\|Tv\|=\|v\|$ for all $v$. Over $\mathbb{R}$ such a $T$ is called orthogonal; over $\mathbb{C}$ it is called unitary.
For linear $T$ on a finite-dimensional inner product space, the following are equivalent:
$A\in M_n(\mathbb{R})$ is orthogonal if $A^tA=I$; $A\in M_n(\mathbb{C})$ is unitary if $A^*A=I$. Equivalently the columns form an orthonormal basis — and since $A^*A=I$ gives $AA^*=I$, so do the rows.
If $T$ is unitary and $Tv=\lambda v$ with $v\ne0$, then $|\lambda|=1$.
$\|v\|=\|Tv\|=\|\lambda v\|=|\lambda|\,\|v\|$, and $\|v\|\ne0$.
Consequently $|\det T|=1$: the determinant is the product of the eigenvalues over $\mathbb{C}$. For a real orthogonal matrix $\det T=\pm1$.
$U=\left(\begin{smallmatrix}0&i\\i&0\end{smallmatrix}\right)$ satisfies $U^*U=I$, so it is unitary. Its entries are not real, so it is not an orthogonal matrix; its eigenvalues are $\pm i$, both of modulus $1$.
Products and inverses of unitary operators are unitary, so they form a group $U(n)$ under composition; the real case gives $O(n)$. The elements of determinant $1$ form the subgroups $SU(n)$ and $SO(n)$.
If $S^*S=I$ and $T^*T=I$ then $(ST)^*(ST)=T^*S^*ST=T^*T=I$; and $(T^{-1})^*T^{-1}=(T^*)^*T^*=TT^*=I$.
If $\beta$ and $\gamma$ are both orthonormal bases, the change-of-basis matrix from $\beta$ to $\gamma$ is unitary. This is why "orthogonally diagonalizable" in Unit 6 means $A=UDU^*$ rather than $A=PDP^{-1}$ — the conjugating matrix can be taken unitary, and then $P^{-1}$ costs nothing to compute.
$A$ and $B$ are unitarily equivalent if $B=U^*AU$ for some unitary $U$. This is stronger than similarity: it preserves the inner product structure as well as the operator. Unitarily equivalent matrices have the same eigenvalues and the same $\operatorname{tr}(A^*A)$.
• Calling a matrix orthogonal because its columns are pairwise
orthogonal. They must also be unit vectors; "orthogonal matrix"
really means orthonormal columns.
• Using $A^t$ instead of $A^*$ over $\mathbb{C}$. The transpose alone does
not give the inverse of a unitary matrix.
• Concluding $\det T=1$. It is $\pm1$ in the real case and any unit
complex number in general.
• Assuming an isometry is diagonalizable over $\mathbb{R}$. A rotation by
$\theta\ne0,\pi$ has no real eigenvalue at all.
Original problems written for this guide in the style of Friedberg, Insel & Spence exercises. Write a full solution before opening the answer.
Is $\tfrac{1}{5}\left(\begin{smallmatrix}3&-4\\4&3\end{smallmatrix} \right)$ orthogonal?
Describe the map with matrix $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$.
Prove every eigenvalue of a unitary operator has $|\lambda|=1$.
Show the product of two unitary matrices is unitary.
Show an isometry maps orthogonal vectors to orthogonal vectors.
For a unit vector $u$, show $T(v)=v-2\langle v,u\rangle u$ is orthogonal and describe it.
Show $R_{\pi/2}$ has no real eigenvalue but is orthogonal.
Show $\operatorname{tr}(A^*A)$ is unchanged under $A\mapsto U^*AU$ for unitary $U$.
Course coverage. This unit is part of the complete MAT247H1 topic map. It should cover the following items before you move on:
Use the current UofT MAT247H1 calendar topics as the coverage spine, and pair lecture notes with a Friedberg-Insel-Spence-style linear algebra text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.
Length preserved ⇒ inner products preserved ⇒ $T^*T=I$ ⇒ orthonormal columns. Each arrow is one short computation, and the chain is the whole unit.
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
State the matrix condition for unitarity.
What values can $\det$ take for a real orthogonal matrix?
Which step of the equivalence proof is not immediate?
How do they differ?