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A matrix $\mathbf{W}_h$ can be formed by concatenating a series of $M$ matrices or vectors, denoted as $\mathbf{W}_h^1, \mathbf{W}_h^2, \dots, \mathbf{W}_h^M$. The mathematical representation of this operation is $\mathbf{W}_h = [\mathbf{W}_h^1 \mathbf{W}_h^2 \dots \mathbf{W}_h^M]$.

Concatenated Weight Matrix Formula

Two matrices, $\mathbf{A}$ and $\mathbf{B}$, are to be combined into a single matrix $\mathbf{W}$ using the operation $\mathbf{W} = [\mathbf{A} \mathbf{B}]$. What specific condition must be true about the dimensions of $\mathbf{A}$ and $\mathbf{B}$ for this operation to be valid, and why is this condition necessary?

Conditions for Matrix Concatenation

A researcher constructs a new matrix, $\mathbf{W}$, by concatenating two existing matrices, $\mathbf{W}^1$ and $\mathbf{W}^2$, such that $\mathbf{W} = [\mathbf{W}^1 \mathbf{W}^2]$. If $\mathbf{W}^1$ has dimensions 4x3 (4 rows, 3 columns) and $\mathbf{W}^2$ has dimensions 4x5 (4 rows, 5 columns), then the resulting matrix $\mathbf{W}$ will have dimensions 8x8.

Consider two matrices, $\mathbf{W}^1 = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\mathbf{W}^2 = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. What is the resulting matrix $\mathbf{W}$ if it is formed by the operation $\mathbf{W} = [\mathbf{W}^1 \mathbf{W}^2]$?

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