How can a rank 1 complex matrix be written as a product of two matrices?

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In summary, a rank 1 complex matrix is a square matrix that has only one independent row or column, with all other rows or columns being linearly dependent. Writing a matrix as a product of two matrices means breaking it down into two separate matrices that, when multiplied, result in the original matrix. This is useful for performing operations and gaining insights into the structure and properties of the matrix. To write a rank 1 complex matrix as a product of two matrices, one must find a vector and its transpose that will serve as the two matrices in the product. Applications of this technique can be found in fields such as linear algebra, signal processing, statistics, data compression, and solving systems of linear equations.
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Euge
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If $M$ is a complex $m \times n$ matrix of rank $1$, show that $M$ can be written as $\bf{uv^T}$ where $\bf{u}$ is an $m\times 1$ matrix and $\bf{v}$ is an $n\times 1$ matrix.

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Congratulations to Opalg and castor28 for their correct solutions. Here is castor28's solution.
As the column space of $M$ has dimension $1$, it is spanned by a single vector $\mathbf{u}$. Therefore, for all $i$, the column $i$ of $M$ can be written as $\mathbf{u}v_i$ for some scalar $v_i$.

This shows that $M=\mathbf{uv^T}$, where $\mathbf{v^T}$ is the column vector $(v_i)$.
 

FAQ: How can a rank 1 complex matrix be written as a product of two matrices?

What is a rank 1 complex matrix?

A rank 1 complex matrix is a square matrix where all the rows and columns are linearly dependent, meaning that one row or column can be expressed as a scalar multiple of another. This results in only one independent row or column, giving the matrix a rank of 1.

What does it mean to write a matrix as a product of two matrices?

Writing a matrix as a product of two matrices means breaking down a given matrix into two separate matrices that, when multiplied together, result in the original matrix. This is similar to factoring a number into its prime factors.

Why is it useful to write a rank 1 complex matrix as a product of two matrices?

Writing a rank 1 complex matrix as a product of two matrices can make it easier to perform operations such as matrix multiplication and finding determinants. It can also provide deeper insights into the structure and properties of the given matrix.

How do you write a rank 1 complex matrix as a product of two matrices?

To write a rank 1 complex matrix as a product of two matrices, you need to find a vector that when multiplied by its transpose (to create a matrix) results in the original matrix. This vector and its transpose matrix will be the two matrices in the product.

What are the applications of writing a rank 1 complex matrix as a product of two matrices?

Writing a rank 1 complex matrix as a product of two matrices has various applications in fields such as linear algebra, signal processing, and statistics. It can also be used in data compression techniques and in solving systems of linear equations.

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