(a) Prove that the product A=𝐯 𝐰^T of a nonzero m ×1 column vector 𝐯 and a nonzero 1 ×n row vect

step 1 Hello, so we consider n by n matrices, A, B, C, and D, such that the rank of A is equal to the r

(a) Prove that the product A=𝐯 𝐰^T of a nonzero m ×1 column...

(a) Prove that the product $A=\mathbf{v} \mathbf{w}^T$ of a nonzero $m \times 1$ column vector $\mathbf{v}$ and a nonzero $1 \times n$ row vector $\mathbf{w}^T$ is an $m \times n$ matrix of rank $r=1$. (b) Compute the following rank one products: (i) $\left(\begin{array}{l}1 \\ 3\end{array}\right)\left(\begin{array}{ll}-1 & 2\end{array}\right)$, (ii) $\left(\begin{array}{r}4 \\ 0 \\ -2\end{array}\right)\left(\begin{array}{ll}-2 & 1\end{array}\right)$, (iii) $\left(\begin{array}{r}2 \\ -3\end{array}\right)\left(\begin{array}{lll}1 & 3 & -1\end{array}\right)$. (c) Prove that every rank one matrix can be written in the form $A=\mathbf{v} \mathbf{w}^T$.

(a) Prove that the product $A=\mathbf{v} \mathbf{w}^T$ of a nonzero $m \times 1$ column vector $\mathbf{v}$ and a nonzero $1 \times n$ row vector $\mathbf{w}^T$ is an $m \times n$ matrix of rank $r=1$. (b) Compute the following rank one products: (i) $\left(\begin{array}{l}1 \\ 3\end{array}\right)\left(\begin{array}{ll}-1 & 2\end{array}\right)$,
(ii) $\left(\begin{array}{r}4 \\ 0 \\ -2\end{array}\right)\left(\begin{array}{ll}-2 & 1\end{array}\right)$,
(iii) $\left(\begin{array}{r}2 \\ -3\end{array}\right)\left(\begin{array}{lll}1 & 3 & -1\end{array}\right)$.
(c) Prove that every rank one matrix can be written in the form $A=\mathbf{v} \mathbf{w}^T$.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Outer Product

The outer product is a fundamental operation in linear algebra where a column vector is multiplied by a row vector, resulting in a matrix. This matrix is constructed such that each element is the product of an element from the column vector and an element from the row vector. It is particularly significant because the resulting matrix always has a rank of one, provided that the original vectors are nonzero.

Matrix Rank and Rank-One Matrices

The rank of a matrix is defined as the maximum number of linearly independent rows or columns it contains. A rank-one matrix is a special case where this number is one, meaning that every row or column is a scalar multiple of a single vector. This property underscores the inherent simplicity of rank-one matrices and their applicability in various areas such as dimensionality reduction and signal processing.

Representation of Rank-One Matrices

A key concept in linear algebra is that any rank-one matrix can be expressed as the outer product of two nonzero vectors. This representation is not only mathematically elegant but also practically useful as it simplifies the analysis and computation involving such matrices, providing insight into their structure and the linear dependencies among their rows and columns.

Transcript

Please give Ace some feedback