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Chapter 7: Problem 3

(a) Let \(A=\left(\begin{array}{rrr}1 & 2 & 0 \\ 3 & -1 & 4\end{array}\right)\). Find \(A A^{T}\) and \(A^{T} A\) and their traces. (b) Let \(M\) be any \(m \times n\) matrix. Show that \(M^{T} M\) and \(M M^{T}\) are symmetric. (Hint: use the result of the previous problem.) What are their sizes? What is the relationship between their traces?

Short Answer

Expert verified
For (a), traces of both matrices are 31. For (b), both matrices are symmetric, \(M^{T}M\) is \(n\times n\), \(M M^{T}\) is \(m\times m\), and they have equal traces.

Step by step solution

01

Calculate the transpose of Matrix A

The transpose of a matrix is obtained by swapping its rows with columns. Given matrix \[ A = \begin{pmatrix} 1 & 2 & 0 \ 3 & -1 & 4 \end{pmatrix}, \]its transpose is \[ A^{T} = \begin{pmatrix} 1 & 3 \ 2 & -1 \ 0 & 4 \end{pmatrix}. \]
02

Calculate AA^T

To calculate \( AA^T \), multiply matrix \( A \) by its transpose \( A^{T} \).\[ A A^{T} = \begin{pmatrix} 1 & 2 & 0 \ 3 & -1 & 4 \end{pmatrix} \begin{pmatrix} 1 & 3 \ 2 & -1 \ 0 & 4 \end{pmatrix} = \begin{pmatrix} (1 \cdot 1 + 2 \cdot 2 + 0 \cdot 0) & (1 \cdot 3 + 2 \cdot (-1) + 0 \cdot 4) \ (3 \cdot 1 + (-1) \cdot 2 + 4 \cdot 0) & (3 \cdot 3 + (-1) \cdot (-1) + 4 \cdot 4) \end{pmatrix} \]\[ = \begin{pmatrix} 5 & 1 \ 1 & 26 \end{pmatrix}. \]
03

Calculate A^TA

To calculate \( A^{T}A \), multiply matrix \( A^{T} \) by \( A \).\[ A^{T} A = \begin{pmatrix} 1 & 3 \ 2 & -1 \ 0 & 4 \end{pmatrix} \begin{pmatrix} 1 & 2 & 0 \ 3 & -1 & 4 \end{pmatrix} = \begin{pmatrix} (1 \cdot 1 + 3 \cdot 3) & (1 \cdot 2 + 3 \cdot (-1)) & (1 \cdot 0 + 3 \cdot 4) \ (2 \cdot 1 + (-1) \cdot 3) & (2 \cdot 2 + (-1) \cdot (-1)) & (2 \cdot 0 + (-1) \cdot 4) \ (0 \cdot 1 + 4 \cdot 3) & (0 \cdot 2 + 4 \cdot (-1)) & (0 \cdot 0 + 4 \cdot 4) \end{pmatrix} \]\[ = \begin{pmatrix} 10 & -1 & 12 \ -1 & 5 & -4 \ 12 & -4 & 16 \end{pmatrix}. \]
04

Calculate the traces of AA^T and A^TA

The trace of a matrix is the sum of its diagonal elements.For \( A A^{T} = \begin{pmatrix} 5 & 1 \ 1 & 26 \end{pmatrix} \), the trace is \[ \text{Tr}(AA^{T}) = 5 + 26 = 31. \]For \( A^{T} A = \begin{pmatrix} 10 & -1 & 12 \ -1 & 5 & -4 \ 12 & -4 & 16 \end{pmatrix} \), the trace is \[ \text{Tr}(A^{T}A) = 10 + 5 + 16 = 31. \]
05

Show MTM and MMT are symmetric

A matrix \( X \) is symmetric if \( X = X^{T} \).For any matrix \( M \), \[ M^{T}M \Rightarrow (M^{T}M)^{T} = M^{T}(M^{T})^{T} = M^{T}M, \]which shows that \( M^{T}M \) is symmetric.Similarly, \[ M M^{T} \Rightarrow (M M^{T})^{T} = (M^{T})^{T} M^{T} = M M^{T}, \]which shows that \( M M^{T} \) is also symmetric.
06

Determine sizes and relationship between traces

The matrix \( M^{T}M \) will be \( n \times n \) and the matrix \( M M^{T} \) will be \( m \times m \). As shown in part (a), for a specific case:- \( \text{Tr}(M^{T} M) = \text{Tr}(M M^{T}) \),which states that both traces are equal, regardless of their sizes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transpose of a Matrix
The transpose of a matrix is a fundamental concept in linear algebra. It involves flipping the matrix over its diagonal. This turns the matrix's rows into columns and vice versa. The notation for the transpose of a matrix is usually a superscript \(^T\). For instance, given a matrix \(A\), its transpose is written as \(A^{T}\).

Let's consider a simple example to clarify this concept. If matrix \(A\) is given by: \[ A = \begin{pmatrix} 1 & 2 & 0 \ 3 & -1 & 4 \end{pmatrix}, \]then the transpose, \(A^{T}\), would be: \[ A^{T} = \begin{pmatrix} 1 & 3 \ 2 & -1 \ 0 & 4 \end{pmatrix}. \]Here are some key points about matrix transpose:
  • The transpose of a transpose returns the original matrix, i.e., \((A^{T})^{T} = A\).
  • The transpose of a product of matrices is the product of their transposes in reverse order, \((AB)^{T} = B^{T}A^{T}\).
Trace of a Matrix
The trace of a matrix is another fundamental idea in the realm of matrices. It is defined as the sum of the elements on the main diagonal of a square matrix. The main diagonal refers to the diagonal that starts from the top left and ends at the bottom right.

For example, if you have a square matrix \(B\): \[ B = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}, \]the trace, denoted by \(\text{Tr}(B)\), would be: \[ \text{Tr}(B) = a + e + i. \]Some notable properties of the trace include:
  • It is only defined for square matrices.
  • The trace of a sum of matrices is the sum of their traces, i.e., \(\text{Tr}(A + B) = \text{Tr}(A) + \text{Tr}(B)\).
  • For matrices \(A\) and \(B\) where \(AB\) is square, \(\text{Tr}(AB) = \text{Tr}(BA)\).
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Most popular questions from this chapter

An example of an operation which is not associative is the cross product. (a) Give a simple example of three vectors from 3-space \(u, v, w\) such that \(u \times(v \times w) \neq(u \times v) \times w\) (b) We saw in Chapter 1 that the operator \(B=u \times(\) cross product with a vector) is a linear operator. It can therefore be written as a matrix (given an ordered basis such as the standard basis). How is it that composing such linear operators is non-associative even though matrix multiplication is associative?

Let \(M=\left(\begin{array}{cccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3\end{array}\right) .\) Divide \(M\) into named blocks, with one block the \(4 \times 4\) identity matrix, and then multiply blocks to compute \(M^{2}\).

Argue that if \(M\) is symmetric then \(L=U^{T}\) in the \(L D U\) decomposition of \(M\)

Find the "matrix" for \(\frac{d}{d x}\) acting on the vector space of all power series in the ordered basis \(\left(1, x, x^{2}, x^{3}, \ldots\right) .\) Use this matrix to find all power series solutions to the differential equation \(\frac{d}{d x} f(x)=x\). Hint: your "matrix" may not have finite size.

Let \(M=\left(\begin{array}{cc}X & Y \\ Z & W\end{array}\right)\) be a square \(n \times n\) block matrix with \(W\) invertible. i. If \(W\) has \(r\) rows, what size are \(X, Y\), and \(Z ?\) ii. Find a \(U D L\) decomposition for \(M\). In other words, fill in the stars in the following equation: $$ \left(\begin{array}{cc} X & Y \\ Z & W \end{array}\right)=\left(\begin{array}{ll} I & * \\ 0 & I \end{array}\right)\left(\begin{array}{ll} * & 0 \\ 0 & * \end{array}\right)\left(\begin{array}{ll} I & 0 \\ * & I \end{array}\right) $$
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