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Unitary matrix

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A unitary matrix is a complex square matrix whose columns (and rows) are orthonormal. It has the remarkable property that its inverse is equal to its conjugate transpose.

A unitary matrix whose entries are all real numbers is said to be orthogonal.

Table of Contents

Preliminary notions

In order to understand the definition of a unitary matrix, we need to remember the following things.

We say that two vectors $r$ and $s$ are orthogonal if and only if their inner product is equal to zero:[eq1]

We can use the inner product to define the norm (length) of a vector $s$ as follows:[eq2]

We say that a set of vectors [eq3] is orthonormal if and only if[eq4]that is, if and only if the elements of the set have unit norm and are orthogonal to each other.

When the vectors are arrays of complex numbers and, in particular, Kx1 column vectors having complex entries, the usual way to define the inner product is[eq5]where $s$ and $r$ are Kx1 vectors and $s^{st }$ denotes the conjugate transpose of $s$.

When the vectors are arrays of real numbers, the inner product is the usual dot product between two vectors:[eq6]where $s^{ op }$ denotes the transpose of $s$.

Definition

We are now ready to give a definition of unitary matrix.

Definition A $K imes K$ complex matrix A is said to be unitary if and only if it is invertible and its inverse is equal to its conjugate transpose, that is,[eq7]

Remember that $A^{-1}$ is the inverse of a $K imes K$ matrix A if and only if it satisfies[eq8]where I is the $K imes K$ identity matrix. As a consequence, the following two propositions hold.

Proposition A is a unitary matrix if and only if[eq9]

Proposition A is a unitary matrix if and only if[eq10]

Let us make a simple example.

Example Define the $2 imes 2$ complex matrix[eq11]The conjugate transpose of A is[eq12]The matrix product between $A^{st }$ and A is[eq13]Then, A is unitary.

Orthonormality of the columns

Unitary matrices have the property that their columns are orthonormal.

Proposition A matrix A is unitary if and only if its columns form an orthonormal set.

Proof

Note that the $left( j,k ight) $-th entry of the identity matrix is [eq14]Moreover, by the very definition of matrix product, the $left( j,k ight) $-th entry of the product $A^{st }A$ is the product between the $j$-th row of $A^{st }$ (denoted by [eq15]) and the $k $-th column of A (denoted by $A_{ullet k}$): [eq16]In turn, by the definition of conjugate transpose, the $j$-th row of $A^{st }$ is equal to the conjugate transpose of the $j$-th column of $A $. Therefore, we have that[eq17]Having established these facts, let us prove the "if" part of the proposition. Suppose that the columns of A form an orthonormal set. Then, [eq18]which implies[eq19]for any $j$ and k. As a consequence, [eq20]which means that A is unitary. Let us now prove the "only if" part. Suppose that A is unitary. Then,[eq21]which implies[eq22]As a consequence, the columns of A are orthonormal.

Example Consider again the matrix[eq23]and denote its two columns by [eq24]The two columns have unit norm because[eq25]and[eq26]They are orthogonal because[eq27]

Unitary transpose

A very simple property follows.

Proposition A matrix A is unitary if and only if its transpose $A^{ op }$ is unitary.

Proof

We already know that A is unitary if and only[eq28]We can transpose both sides of the equation and obtain the equivalent condition[eq29]where we have used the fact that the order of conjugation and transposition does not matter. The latter condition is satisfied if and only if $A^{ op }$ is unitary, which proves the proposition.

Orthonormality of the rows

Not only the columns but also the rows of a unitary matrix are orthonormal.

Proposition A matrix A is unitary if and only if its rows form an orthonormal set.

Proof

The rows of A are the columns of $A^{ op }$, which is unitary 1) if and only if it has orthonormal columns; 2) if and only if A is unitary.

Conjugate transpose

Another proposition that can be proved in few lines.

Proposition A matrix A is unitary if and only if its conjugate transpose $A^{st }$ is unitary.

Proof

We already know that A is unitary if and only if [eq29]By taking the complex conjugate of both sides of the equation, we obtain[eq31]or[eq32]which is equivalent to saying that $A^{st }$ is unitary.

Product of unitary matrices

The product of unitary matrices is a unitary matrix.

Proposition Let A and $B$ be two unitary $K imes K$ matrices. Then, the product $AB$ is unitary.

Proof

The conjugate transpose of $AB$ is[eq33]Therefore,[eq34]which implies that $AB$ is unitary.

Unitary and triangular matrices

The following fact is sometimes used in matrix algebra.

Proposition Let A be a $K imes K$ unitary matrix. If A is triangular (either lower or upper) and its diagonal entries are positive, then[eq35]where I is the $K imes K$ identity matrix.

Proof

Let us start from the case in which A is upper triangular (UT). Since A is UT, only the first entry of its first column can be different from zero:[eq36]Since A is unitary, the norm of one of its columns must be equal to 1. Since by assumption the diagonal entries of A must be positive, the norm of the column is unitary only if[eq37]that is, if $A_{ullet 1}$ is the first vector of the canonical basis. Since A is UT, only the first two entries of its second column can be different from zero:[eq38]The inner product between the first two columns of A is[eq39]Since the columns of A are orthogonal to each other, the latter inner product must be equal to zero, which implies that [eq40]Therefore,[eq41]Since the norm of $A_{ullet 2}$ must be equal to 1, it must be that $A_{22}=1$. Thus, $A_{ullet 2}$ is the second vector of the canonical basis. For each of the other columns of A, we proceed similarly: we impose that some entries of the column be equal to zero because A is triangular; we prove that other entries must be equal to zero in order to satisfy the orthogonality conditions; we prove that the only non-zero entry must be equal to 1 in order to satisfy the requirement of normality. The process ends when we have proved that $A_{ullet k}$ is equal to the k-th column of the canonical basis, for $k=1,ldots ,K$. Thus, A is equal to the $K imes K$ identity matrix. If A is lower triangular and unitary, then $A^{ op }$ is upper triangular and unitary. As a consequence, we have that $A^{ op }=I$, which implies that $A=I$.

Non-square matrices with orthonormal columns

The most important property of unitary matrices applies also to matrices that are not square but have orthonormal columns.

Proposition Let A be a $K imes L$ matrix such that its $L$ columns form an orthonormal set. Then, [eq42]where I is the $L imes L$ identity matrix.

Proof

Denote by $A_{ullet l}$ the $l$-th column of A. By the definition of matrix product, the matrix[eq43]is an $L imes L$ matrix whose $left( l,m ight) $-th entry is[eq44]because the columns of A are orthonormal. In other words,[eq45]where I is the $L imes L$ identity matrix.

Orthogonal matrix

If all the entries of a unitary matrix are real (i.e., their complex parts are all zero), then the matrix is said to be orthogonal.

If A is a real matrix, it remains unaffected by complex conjugation. As a consequence, we have that[eq46]

Therefore a real matrix is orthogonal if and only if[eq47]

Since an orthogonal matrix is unitary, all the properties of unitary matrices apply to orthogonal matrices.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Define the matrix[eq48]

Find a scalar $lpha $ such that $lpha A$ is unitary.

Solution

We need to find $lpha $ such that[eq49]Let us first compute the conjugate transpose of A:[eq50]Then, we can compute its product with A:[eq51]Thus, if we choose $lpha =1/2$, we obtain[eq52]

How to cite

Please cite as:

Taboga, Marco (2021). "Unitary matrix", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/unitary-matrix.

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