# Mianzhi Wang

Ph.D. in Electrical Engineering

# Vectorization, Kronecker Product, and Khatri-Rao Product

In array and radar signal processing, especially when co-array models are concerned, one may frequently encounter the vectorization operation, the Kronecker product, and the Khatri-Rao product. This article will give a brief review of these three operations and their commonly used properties.

## Vectorization

Given an $M \times N$ matrix $\mathbf{A} = [\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_N]$, where $\mathbf{a}_i$ is the $i$-th column of $A$. The vectorization of $\mathbf{A}$ is defined as

$\mathrm{vec}(\mathbf{A}) = [\mathbf{a}_1^T, \mathbf{a}_2^T, \ldots, \mathbf{a}_N^T]^T.$
(1)

Basically, the vectorization operation rearranges the elements of $\mathbf{A}$ into a long vector by stacking all the columns together. In a general signal processing scenario, we may have $N$ observations, $\mathbf{x}_1$, $\mathbf{x}_2$, ..., $\mathbf{x}_n$. We can either arrange them into a matrix $\mathbf{X} = [\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N]$, or a long vector $\mathrm{vec}(\mathbf{X})$, depending on the problem structure. In some cases, vectorizing a matrix may be helpful. However, the additional information given by the matrix structure (e.g., low-rankness) will be lost.

The vectorization operation itself does not have many interesting properties. One of the useful ones is related to trace:

$\mathrm{tr}(\mathbf{A}\mathbf{B}) = \mathrm{vec}(\mathbf{A}^T)^T \mathrm{vec}(\mathbf{B}) = \mathrm{vec}(\mathbf{A}^H)^H \mathrm{vec}(\mathbf{B}).$
(2)

As we will show later, the vectorization operation shows more interesting properties when combined with the Kronecker product and the Khatri-Rao product.

## Kronecker Product

Given an $M \times N$ matrix $A$ and a $P \times Q$ matrix B, the Kronecker product $\mathbf{A}\otimes\mathbf{B}$ is defined as

$\mathbf{A}\otimes\mathbf{B} = \begin{bmatrix} a_{11}\mathbf{B} & a_{12}\mathbf{B} & \cdots & a_{1N}\mathbf{B}\\ a_{21}\mathbf{B} & a_{22}\mathbf{B} & \cdots & a_{2N}\mathbf{B}\\ \vdots & \vdots & \ddots & \vdots \\ a_{M1}\mathbf{B} & a_{M2}\mathbf{B} & \cdots & a_{MN}\mathbf{B} \end{bmatrix},$
(3)

which is an $MP \times NQ$ matrix.

The Kronecker product has many interesting properties. First, it is distributive and associative:

• Distributivity: (a) $(\mathbf{A} + \mathbf{B}) \otimes \mathbf{C} = \mathbf{A} \otimes \mathbf{C} + \mathbf{B} \otimes \mathbf{C}$; (b) $\mathbf{A} \otimes (\mathbf{B} + \mathbf{C}) = \mathbf{A} \otimes \mathbf{B} + \mathbf{A} \otimes \mathbf{C}$.
• Associativity: $(\mathbf{A} \otimes \mathbf{B}) \otimes \mathbf{C} = \mathbf{A} \otimes (\mathbf{B} \otimes \mathbf{C})$.

The Kronecker product is also "distributive" with respect to the (Hermitian) transpose, trace, and Frobenius norm:

• Transpose: $(\mathbf{A} \otimes \mathbf{B})^T = \mathbf{A}^T \otimes \mathbf{B}^T$.
• Hermitian transpose: $(\mathbf{A} \otimes \mathbf{B})^H = \mathbf{A}^H \otimes \mathbf{B}^H$.
• Trace: $\mathrm{tr}(\mathbf{A} \otimes \mathbf{B}) = \mathrm{tr}(\mathbf{A}) \mathrm{tr}(\mathbf{B})$.
• Frobenius norm: $\|\mathbf{A} \otimes \mathbf{B}\|_F = \|\mathbf{A}\|_F \|\mathbf{B}\|_F$.

One of the most important and useful properties of the Kronecker product is the product rule:

Proposition 1. Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ be $M \times N$, $P \times Q$, $N \times K$, and $Q \times L$, respectively, then

$(\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \otimes \mathbf{D}) = (\mathbf{A}\mathbf{C}) \otimes (\mathbf{B}\mathbf{D}).$
(4)

Proof. By the definition of the Kronecker product,

\begin{aligned} &(\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \otimes \mathbf{D})\\ =&\begin{bmatrix} a_{11}\mathbf{B} & a_{12}\mathbf{B} & \cdots & a_{1N}\mathbf{B}\\ a_{21}\mathbf{B} & a_{22}\mathbf{B} & \cdots & a_{2N}\mathbf{B}\\ \vdots & \vdots & \ddots & \vdots \\ a_{M1}\mathbf{B} & a_{M2}\mathbf{B} & \cdots & a_{MN}\mathbf{B} \end{bmatrix} \begin{bmatrix} c_{11}\mathbf{D} & c_{12}\mathbf{D} & \cdots & c_{1K}\mathbf{D}\\ c_{21}\mathbf{D} & c_{22}\mathbf{D} & \cdots & c_{2K}\mathbf{D}\\ \vdots & \vdots & \ddots & \vdots \\ c_{N1}\mathbf{D} & c_{N2}\mathbf{D} & \cdots & c_{NK}\mathbf{D} \end{bmatrix}\\ =& \begin{bmatrix} (\sum_{i=1}^N a_{1i}c_{i1})\mathbf{B}\mathbf{D} & (\sum_{i=1}^N a_{1i}c_{i2})\mathbf{B}\mathbf{D} & \cdots & (\sum_{i=1}^N a_{1i}c_{iK})\mathbf{B}\mathbf{D} \\ (\sum_{i=1}^N a_{2i}c_{i1})\mathbf{B}\mathbf{D} & (\sum_{i=1}^N a_{2i}c_{i2})\mathbf{B}\mathbf{D} & \cdots & (\sum_{i=1}^N a_{2i}c_{iK})\mathbf{B}\mathbf{D} \\ \vdots & \vdots & \ddots & \vdots \\ (\sum_{i=1}^N a_{Mi}c_{i1})\mathbf{B}\mathbf{D} & (\sum_{i=1}^N a_{Mi}c_{i2})\mathbf{B}\mathbf{D} & \cdots & (\sum_{i=1}^N a_{Mi}c_{iK})\mathbf{B}\mathbf{D} \\ \end{bmatrix}\\ =& (\mathbf{A}\mathbf{C}) \otimes (\mathbf{B}\mathbf{D}), \end{aligned}

which completes the proof. ∎

With the product rule, one can show that the following properties also hold:

• Inverse: $(\mathbf{A} \otimes \mathbf{B})^{-1} = \mathbf{A}^{-1} \otimes \mathbf{B}^{-1}$.
• Rank: $\mathrm{rank}(\mathbf{A} \otimes \mathbf{B}) = \mathrm{rank}(\mathbf{A}) \mathrm{rank}(\mathbf{B})$.
• Determinant: Let $\mathbf{A}$ and $\mathbf{B}$ be $M \times M$ and $N \times N$, respectively. Then $\det(\mathbf{A} \otimes \mathbf{B}) = \det(\mathbf{A})^N \det(\mathbf{B})^M$.
• Positive-definiteness： If both $\mathbf{A}$ and $\mathbf{B}$ are positive-definite, then $\mathbf{A} \otimes \mathbf{B}$ is also positive-definite.

Remark 1. The proof for the inversion one is pretty straight-forward because

$(\mathbf{A}^{-1} \otimes \mathbf{B}^{-1})(\mathbf{A} \otimes \mathbf{B}) = (\mathbf{A}^{-1}\mathbf{A}) \otimes (\mathbf{B}^{-1}\mathbf{B}) = \mathbf{I},$

and the other direction also holds. The one about rank can be shown using singular-value decomposition, and the one about positive-definiteness can be shown with eigen-decomposition. The one about determinant is tricker. The key idea is using the following equality: $\mathbf{A} \otimes \mathbf{B} = (\mathbf{A} \otimes \mathbf{I})(\mathbf{I} \otimes \mathbf{B})$.

For a complete review of the properties of the Kronecker product, the readers are directed to the wiki page, Kathrin Schäcke's On the Kronecker Product, or Chapter 11 in A matrix handbook for statisticians. Readers pursuing a more abstract understanding may also check out the tensor product.

## Khatri-Rao Product

The Khatri-Rao product is usually defined as the column-wise Kronecker product. In other words, given an $M \times N$ matrix $\mathbf{A}$ and a $P \times N$ matrix $\mathbf{B}$, the Khatri-Rao product is defined as

$\mathbf{A}\odot\mathbf{B} = \begin{bmatrix} \mathbf{a}_1 \otimes \mathbf{b}_1 & \mathbf{a}_2 \otimes \mathbf{b}_2 & \cdots \mathbf{a}_N \otimes \mathbf{b}_N \end{bmatrix},$
(5)

which is an $MP \times N$ matrix. The Khatri-Rao product appears frequently in the difference co-array model (e.g., for co-prime and nested arrays) or sum-coarray model (e.g., in MIMO radar). Although the definition of the Khatri-Rao product is based on the Kronecker product, the Khatri-Rao product does not have many nice properties.

## A Property That Connects the Three

A handy property connecting the vectorization, the Kronecker product, and the Khatri-Rao product is given below.

Proposition 2. Let $\mathbf{A}$, $\mathbf{X}$, $\mathbf{B}$ be $M \times N$, $N \times P$, $P \times Q$, respectively. Then

$\mathrm{vec}(\mathbf{A}\mathbf{X}\mathbf{B}) = (\mathbf{B}^T \otimes \mathbf{A}) \mathrm{vec}(\mathbf{X}).$
(6)

Moreover, if $\mathbf{X}$ is a diagonal matrix, then

$\mathrm{vec}(\mathbf{A}\mathbf{X}\mathbf{B}) = (\mathbf{B}^T \odot \mathbf{A}) \mathrm{diag}(\mathbf{X}),$
(7)

where $\mathrm{diag}(\mathbf{X})$ is a vector representing the main diagonal of $\mathbf{X}$.

Proof. Let $\mathbf{B}_i$ denote the $i$-th row of $\mathbf{B}$.

\begin{aligned} \mathrm{vec}(\mathbf{A}\mathbf{X}\mathbf{B}) =& \sum_{i=1}^N \sum_{j=1}^P x_{ij} \mathrm{vec}(\mathbf{a}_i \mathbf{B}_j)\\ =& \sum_{i=1}^N \sum_{j=1}^P x_{ij} (\mathbf{B}_j^T \otimes \mathbf{a}_i)\\ =& \sum_{j=1}^P (\mathbf{B}_j^T \otimes \mathbf{A})\mathbf{x}_j\\ =& (\mathbf{B}^T \otimes \mathbf{A}) \mathrm{vec}(\mathbf{X}) \end{aligned}

The proof for the second equality follows the same idea. ∎

Here are some examples where Proposition 2 is used.

Example 1. Consider the following optimization problem with a matrix variable:

$\min_{\mathbf{X}} \|\mathbf{A} \mathbf{X} - \mathbf{B}\|_F^2.$
(8)

Apply Proposition 2, we can rewrite the above optimization problem as

$\min_{\mathrm{vec}(\mathbf{X})} \| (\mathbf{I} \otimes \mathbf{A}) \mathrm{vec}(\mathbf{X}) - \mathrm{vec}(\mathbf{B}) \|_2^2,$
(9)

which is indeed a least squares problem. Note that the original formulation is more compact.

Example 2. Consider the DOA estimation problem using a linear array. Adapting the unconditional model with uncorrelated sources, the covariance matrix of the received snapshots is given by

$\mathbf{R} = \mathbf{A}\mathbf{P}\mathbf{A}^H + \sigma^2\mathbf{I},$
(10)

where $\mathbf{A}$ is the array steering matrix, $\mathbf{P}$ is a diagonal matrix whose main diagonal represents source powers, and $\sigma^2$ is the power of additive noises. Vectorizing $\mathbf{R}$ leads to

$\mathrm{vec}(\mathbf{R}) = (\mathbf{A}^* \odot \mathbf{A}) \mathbf{p} + \sigma^2\mathrm{vec}(\mathbf{I}),$
(11)

where $\mathbf{p} = \mathrm{diag}(\mathbf{P})$. We can observe that (11) resembles a snapshot from a virtual array whose steering matrix is given by $(\mathbf{A}^* \odot \mathbf{A})$. This idea is exploited to obtain enhanced degrees of freedom.

Proposition 2 also leads to the following interesting corollary:

Corollary 1. Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ be $M \times N$, $N \times P$, $P \times Q$, and $Q \times R$, respectively. Then

$\mathrm{tr}(\mathbf{A}\mathbf{B}\mathbf{C}\mathbf{D}) = \mathrm{vec}(\mathbf{A}^T)^T (\mathbf{D}^T \otimes \mathbf{B}) \mathrm{vec}(\mathbf{C}).$
(12)

Proof. Immediately obtained by combining Proposition 2 and the fact that $\mathrm{tr}(\mathbf{A}\mathbf{B}) = \mathrm{vec}(\mathbf{A}^T)^T \mathrm{vec}(\mathbf{B})$. ∎

Corollary 1 can be quite useful in the derivation of the Cramér-Rao bound (CRB) for Gaussian models, as shown in the following example.

Example 3. Let $\mathbf{x}$ follow the complex circularly-symmetric Gaussian distribution $\mathcal{CN}(\mathbf{0}, \mathbf{R}(\mathbf{\theta}))$, where $\mathbf{\theta} \in \mathbb{R}^K$ denotes the parameters to be estimated. The $(i,j)$-th element of the Fisher information matrix (FIM) is then given by

$\mathrm{FIM}_{ij} = \mathrm{tr}\left(\frac{\partial\mathbf{R}}{\partial\theta_i}\mathbf{R}^{-1} \frac{\partial\mathbf{R}}{\partial\theta_j}\mathbf{R}^{-1}\right).$
(13)

In many cases, evaluating each element according the above formula can be quite tedious. Let $\mathbf{r} = \mathrm{vec}(\mathbf{R})$. Using Corollary 1 and the fact the $\mathbf{R}$ is Hermitian, we can rewrite the above formula as

$\mathrm{FIM}_{ij} = \left[\frac{\partial \mathbf{r}}{\partial\theta_i}\right]^H (\mathbf{R}^T \otimes \mathbf{R})^{-1} \frac{\partial \mathbf{r}}{\partial\theta_j}.$
(14)

Let

$\frac{\partial \mathbf{r}}{\partial\mathbf{\theta}} = \left[ \frac{\partial \mathbf{r}}{\partial\theta_1}\ \frac{\partial \mathbf{r}}{\partial\theta_2}\ \cdots\ \frac{\partial \mathbf{r}}{\partial\theta_K} \right].$

We immediate obtain a compact representation of the FIM:

$\mathrm{FIM} = \left[\frac{\partial \mathbf{r}}{\partial\mathbf{\theta}}\right]^H (\mathbf{R}^T \otimes \mathbf{R})^{-1} \frac{\partial \mathbf{r}}{\partial\mathbf{\theta}}.$
(15)

In some cases, this may simplify the derivation of the corresponding CRB.

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