transpose of a vector – Problems in Mathematics https://yutsumura.com Tue, 26 Dec 2017 09:05:00 +0000 en-US hourly 1 https://wordpress.org/?v=5.3.6 https://i2.wp.com/yutsumura.com/wp-content/uploads/2016/12/cropped-question-logo.jpg?fit=32%2C32&ssl=1 transpose of a vector – Problems in Mathematics https://yutsumura.com 32 32 114989322 Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$ https://yutsumura.com/prove-that-mathbfv-mathbfvtrans-is-a-symmetric-matrix-for-any-vector-mathbfv/ https://yutsumura.com/prove-that-mathbfv-mathbfvtrans-is-a-symmetric-matrix-for-any-vector-mathbfv/#respond Mon, 25 Dec 2017 02:13:56 +0000 https://yutsumura.com/?p=6301 Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix.   Definition (Symmetric Matrix). A matrix $A$ is called symmetric if $A^{\trans}=A$. In terms of entries, an...

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]]> Problem 640

Let $\mathbf{v}$ be an $n \times 1$ column vector.

Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix.

 

Definition (Symmetric Matrix).

A matrix $A$ is called symmetric if $A^{\trans}=A$.

In terms of entries, an $n\times n$ matrix $A=(a_{ij})$ is symmetric if $a_{ij}=a_{ji}$ for all $1 \leq i, j \leq n$.

Proof.

Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$. Then we have
\begin{align*}
\mathbf{v} \mathbf{v}^\trans = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}\begin{bmatrix}
v_1 & v_2 & \cdots & v_n
\end{bmatrix}=
\begin{bmatrix} v_1 v_1 & v_1 v_2 & \cdots & v_1 v_n \\ v_2 v_1 & v_2 v_2 & \cdots & v_2 v_n \\ \vdots & \vdots & \vdots & \vdots \\ v_n v_1 & v_n v_2 & \cdots & v_n v_n \end{bmatrix}.
\end{align*}

In particular, the the $(i, j)$-th component is
\[(\mathbf{v} \mathbf{v}^\trans)_{i j} = v_i v_j = v_j v_i = (\mathbf{v} \mathbf{v}^\trans)_{j i}.\] This shows that the matrix $\mathbf{v} \mathbf{v}^\trans$ is symmetric.

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]]> https://yutsumura.com/prove-that-mathbfv-mathbfvtrans-is-a-symmetric-matrix-for-any-vector-mathbfv/feed/ 0 6301 The Length of a Vector is Zero if and only if the Vector is the Zero Vector https://yutsumura.com/the-length-of-a-vector-is-zero-if-and-only-if-the-vector-is-the-zero-vector/ https://yutsumura.com/the-length-of-a-vector-is-zero-if-and-only-if-the-vector-is-the-zero-vector/#respond Mon, 25 Dec 2017 01:56:44 +0000 https://yutsumura.com/?p=6297 Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$.   Proof. Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2...

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]]> Problem 639

Let $\mathbf{v}$ be an $n \times 1$ column vector.

Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$.

 

Proof.

Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} $.

Then we have
\[\mathbf{v}^\trans \mathbf{v} = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} = \sum_{i=1}^n \mathbf{v}_i^2 . \] Because each $v_i^2$ is non-negative, this sum is $0$ if and only if $v_i = 0$ for each $i$. In this case, $\mathbf{v}$ is the zero vector.

Comment.

Recall that the the length of the vector $\mathbf{v}\in \R^n$ is defined to be
\[\|\mathbf{v}\| :=\sqrt{\mathbf{v}^{\trans} \mathbf{v}}.\]

The problem implies that the length of a vector is $0$ if and only if the vector is the zero vector.

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]]> https://yutsumura.com/the-length-of-a-vector-is-zero-if-and-only-if-the-vector-is-the-zero-vector/feed/ 0 6297