Question

An example of a \(3 \times 3\) matrix of rank one is

Short answer

A rank one example is \( \begin{bmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{bmatrix} \).

Step-by-step solution

Understanding Rank One Matrix

A matrix of rank one means that all its rows (or columns) can be represented as multiples of a single vector. Therefore, the matrix has at least one row or column that can generate the others through linear combinations.

Constructing a Matrix from Row and Column Vectors

To construct a rank one matrix, choose a column vector, such as \( \begin{bmatrix} a \ b \ c \end{bmatrix} \), and a row vector, say \( \begin{bmatrix} x & y & z \end{bmatrix} \). The matrix formed by the outer product of these vectors is \( \begin{bmatrix} a \ b \ c \end{bmatrix} \begin{bmatrix} x & y & z \end{bmatrix} = \begin{bmatrix} ax & ay & az \ bx & by & bz \ cx & cy & cz \end{bmatrix} \).

Setting Values for Simplicity

To simplify, we can choose specific values for vectors. For example, let the column vector be \( \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} \) and the row vector be \( \begin{bmatrix} 1 & 1 & 1 \end{bmatrix} \). The resulting matrix is \( \begin{bmatrix} 1 \times 1 & 1 \times 1 & 1 \times 1 \ 2 \times 1 & 2 \times 1 & 2 \times 1 \ 3 \times 1 & 3 \times 1 & 3 \times 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \ 2 & 2 & 2 \ 3 & 3 & 3 \end{bmatrix} \).

Verifying the Rank

Check if the constructed matrix is of rank one by verifying if rows are multiples of each other, or if we can express them as a single vector times a scalar. Indeed, each subsequent row is a scalar multiple of the first row: \( [1 & 1 & 1], [2 & 2 & 2], [3 & 3 & 3]. \) Hence, the rank is one.

Key concepts

Rank One Matrix

A **rank one matrix** is a unique type of matrix where all of the rows (or columns) are linearly dependent. This essentially means that every row or column can be expressed as a scalar multiple of another. In simpler terms, there is one fundamental row or column, and all others in the matrix are just multiples of it.

This property simplifies many calculations and provides a clear pattern within the matrix structure. When dealing with a rank one matrix, it's comforting to know that if you understand one row or column, you understand them all.

Such matrices are often used in data reduction techniques and can be pivotal in simplifying complex linear transformations into more manageable computations.

Linear Combinations

A **linear combination** involves adding together multiple terms, each multiplied by a scalar, to form a new expression. In the case of matrices, specifically rank one matrices, the concept of linear combinations plays a key role.

For example, if we have a fundamental vector, say \(\begin{bmatrix} a \ b \ c \end{bmatrix}\), all other rows or columns in the matrix can be formed using this vector multiplied by different scalars like in \(\begin{bmatrix} ax \ bx \ cx \end{bmatrix}\). This dependency is due to the rank being one, indicating the entire matrix is built through these combinations.

Understanding how linear combinations work can help in recognizing patterns and simplifying matrix-based problems.

Matrix Construction

**Matrix construction** for rank one matrices involves crafting a matrix by selecting one row vector and one column vector then forming their outer product.

The outer product results in a new matrix where each element is the product of elements from the row and column vectors. For instance, take a column vector \(\begin{bmatrix} a \ b \ c \end{bmatrix}\) and a row vector \(\begin{bmatrix} x & y & z \end{bmatrix}\). The resultant matrix would be \(\begin{bmatrix} ax & ay & az \ bx & by & bz \ cx & cy & cz \end{bmatrix}\).

This method makes constructing a rank one matrix straightforward, as it exploits the simplicity of outer products to maintain the rank structure.

Row and Column Vectors

Both **row and column vectors** are fundamental in the study of matrices, especially rank one matrices. A row vector is a 1 × n matrix, which means it's composed of a single row and multiple columns. Contrastingly, a column vector is an n × 1 matrix with a single column and multiple rows.

In constructing rank one matrices, these vectors are paramount. Their outer product forms the matrix that typically contains repeating patterns of rows or columns, reflecting the rank-one nature.

• **Row Vector Example**: A row vector \(\begin{bmatrix} x & y & z \end{bmatrix}\) might be used in combination with a column vector.
• **Column Vector Example**: A column vector \(\begin{bmatrix} a \ b \ c \end{bmatrix}\) can multiply with a row vector to contribute to the entirety of the matrix pattern.

These vectors demonstrate how individual components come together to form structured and rank-specific matrices.