Computing Matrix-Vector Products by Rows

One Row, One Output Coordinate

A matrix-vector product can be calculated one coordinate at a time. The row-by-row method pairs the entries of a row with the entries of the vector in order, multiplies each pair, and adds the results. Each row gives one scalar, and those scalars, in row order, form the product vector. This makes it important to keep track of the matrix’s number of rows and columns as well as the order of every entry.

Throughout, scalars are real. Let \(\mathbf{A}\) be an \(m\times n\) matrix and let \(\mathbf{x}\) be a column vector in \(\mathbb{R}^n\). Write the entries of \(\mathbf{A}\) as \(a_{ij}\), where \(i\) specifies a row and \(j\) specifies a column, and write the entries of \(\mathbf{x}\) as \(x_1,\dots,x_n\). The number of entries in \(\mathbf{x}\) matches the number of entries in each row of \(\mathbf{A}\), so each row can be paired with the whole vector.

For example, if row \(i\) is \((a_{i1},a_{i2},\dots,a_{in})\), its contribution is the single number \(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in}x_n\). The first row produces the first output entry, the second row produces the second, and so on. Thus an \(m\times n\) matrix times a vector with \(n\) entries produces a vector with \(m\) entries. The \(n\) entries tell us how many products are added for each row; the \(m\) rows tell us how many output entries there are.

Proof. By the definition of matrix multiplication, the entry in row \(i\) of the product is obtained by multiplying each entry in row \(i\) of \(\mathbf{A}\) by the corresponding entry of the column vector \(\mathbf{x}\), then summing over the \(n\) columns. That entry is therefore \(\sum_{j=1}^{n}a_{ij}x_j=a_{i1}x_1+\cdots+a_{in}x_n\). Since the product has one such entry for each of the \(m\) rows of \(\mathbf{A}\), it has \(m\) coordinates. \(\square\)

Calculating the Product in Order

A reliable calculation follows the same sequence for each row: write down the row’s products, evaluate them, and add. Keep the row and vector entries in their original order. The product is written as a column vector, with the row results appearing from top to bottom. The examples below include the dimensions so the number of input and output entries can be checked as part of the calculation.

What a Zero Row Does

A row filled with zeros produces a zero output entry, whatever the entries of the input vector are. More generally, if two matrices of the same size have the same row at a given position, multiplying them by the same compatible vector gives the same output at that position. These facts follow directly from the row formula; they are useful checks when carrying out a calculation.

Proof. If row \(i\) of \(\mathbf{A}\) is zero, then \(a_{ij}=0\) for every \(j=1,\dots,n\). The row formula gives the \(i\)th coordinate as \(\sum_{j=1}^{n}a_{ij}x_j=\sum_{j=1}^{n}0x_j=0\). Now suppose row \(i\) of \(\mathbf{A}\) equals row \(i\) of \(\mathbf{B}\). Then \(a_{ij}=b_{ij}\) for every \(j\). Applying the row formula to each product gives \(\sum_{j=1}^{n}a_{ij}x_j=\sum_{j=1}^{n}b_{ij}x_j\), so their \(i\)th output coordinates agree. \(\square\)

Keeping Dimensions and Entry Order Straight

Before multiplying, record the matrix size and the number of vector entries. For an \(m\times n\) matrix and an \(n\)-entry column vector, each row has exactly \(n\) entries to pair with the vector, and there are \(m\) row calculations. This is a practical way to catch a mismatch: an input entry must be available for each column position, and the final column must contain one result per row.

The row index and column index answer different questions. In \(a_{ij}\), \(i\) selects which output coordinate is being calculated, because it selects the row. The index \(j\) selects which pair of entries is multiplied within that row. For instance, \(a_{23}\) is in row 2 and column 3: it contributes to the second output coordinate and is multiplied by \(x_3\). Keeping these roles separate prevents entries from being assigned to the wrong coordinate.

The order within each row matters too. A row \((a_{i1},a_{i2},a_{i3})\) pairs with \((x_1,x_2,x_3)^{\top}\) as \(a_{i1}x_1+a_{i2}x_2+a_{i3}x_3\). Reversing the vector entries changes which values are multiplied together, and can change the answer even when all the same numbers still appear in the calculation.

Common Mistake: Reversing the Vector Entries

Consider the row \((1,4,-2)\) and the vector \(\mathbf{x}=(3,-1,2)^{\top}\). A tempting but incorrect calculation reverses the vector entries while pairing them with the row:

This does not use the entries in corresponding positions: the first row entry must be paired with \(x_1=3\), not with \(x_3=2\). The correct dot product is

Thus the row produces \(-5\), not \(-8\). A useful correction is to write the entries under one another before multiplying: first row entry with first vector entry, second with second, and third with third. Also preserve the output’s column shape: the scalar found from one row is one coordinate, and all row results are placed vertically in their original row order.

When checking an answer, do not merely count its entries. Recalculate each one from its row and compare the resulting scalar with the corresponding coordinate of the proposed product. This checks both the arithmetic and the pairing order. If a row is zero, its output coordinate must be zero; if two rows are identical, their output coordinates must agree. These are quick consistency checks, not replacements for calculating every row.