Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q6E (page 413)

The singular value of the $2 脳 1$ matrix $[\begin{matrix} 3 \\ 4 \end{matrix}]$ is 5.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

Step 1: Definition of singular values:

Consider a matrix $A$ or order $n 脳 m$ . The singular values $蟽_{i}$ of $A$ are the square roots of the eigenvalues of $A^{T} A$ , where $A^{T} A$ is a symmetric matrix and $蟽_{i} 猢� 0$ .

To Find TRUE or FALSE

Let A be the square or non-square matrix.

Compute $A^{T} A$ as follows:

The given matrix is $A = [\begin{matrix} 3 \\ 4 \end{matrix}]$ , then

$A^{T} = [\begin{matrix} 3 & 4 \end{matrix}]$ , and

$A A^{T} = [\begin{matrix} 3 \\ 4 \end{matrix}] [\begin{matrix} 3 & 4 \end{matrix}] A A^{T} = 25$

Find the eigen value of $A^{T} A$ :

It is known that the diagonal entries of a diagonal matrix are eigen values of the matrix.

So, 25 is theeigen value of $A^{T} A$ .

Now, the singular value of A is:

$蟽 = \sqrt{25} = 5$

Hence, the given statement is TRUE.

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91影视

The singular value of the $2 脳 1$ matrix $[\begin{matrix} 3 \\ 4 \end{matrix}]$ is 5.

Short Answer

Step by step solution

Step 1: Definition of singular values:

To Find TRUE or FALSE

One App. One Place for Learning.

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91影视

The singular value of the 2脳1 matrix [34]is 5.

Short Answer

Step by step solution

Step 1: Definition of singular values:

To Find TRUE or FALSE

One App. One Place for Learning.

Most popular questions from this chapter

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The singular value of the $2 脳 1$ matrix $[\begin{matrix} 3 \\ 4 \end{matrix}]$ is 5.