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Chapter 3: Problem 23

The trace \(\operatorname{Tr}(A)\) of the (square) \(n \times n\) matrix \(A\) is defined to be the sum of its diagnonal elements. Prove that, for \(n \times n\) matrices \(A\) and \(B\), we have $$ \operatorname{Tr}(A+B)=\operatorname{Tr}(A)+\operatorname{Tr}(B) \text { and } \operatorname{Tr}(A B)=\operatorname{Tr}(B A) $$ Use these results to deduce that there are no \(n \times n\) matrices \(A\) and \(B\) such that \(A B-B A=I_{n}\)

Short Answer

Expert verified
It's impossible that \(AB - BA = I_n\) because it's equivalent to \(0 = n\), a contradiction.

Step by step solution

01

Understanding Traces

The trace of a matrix, \(\operatorname{Tr}(A)\), is simply the sum of the elements on its main diagonal. For an \(n \times n\) matrix \(A = [a_{ij}]\), the trace is calculated as \(\operatorname{Tr}(A) = \sum_{i=1}^{n} a_{ii}\). This concept is applied to both the sum and product of matrices.
02

Proving \(\operatorname{Tr}(A+B) = \operatorname{Tr}(A) + \operatorname{Tr}(B)\)

To prove this, note that \( (A+B)_{ii} = a_{ii} + b_{ii} \) for the diagonal elements. Therefore, the trace of \(A + B\) is \( \sum_{i=1}^{n}(a_{ii} + b_{ii}) = \sum_{i=1}^{n}a_{ii} + \sum_{i=1}^{n}b_{ii} = \operatorname{Tr}(A) + \operatorname{Tr}(B) \).
03

Proving \(\operatorname{Tr}(AB) = \operatorname{Tr}(BA)\)

Consider \( (AB)_{ii} = \sum_{k=1}^{n} a_{ik}b_{ki} \) and \( (BA)_{ii} = \sum_{k=1}^{n} b_{ik}a_{ki} \). The trace of each is \(\sum_{i=1}^{n}\sum_{k=1}^{n}a_{ik}b_{ki}\) and \(\sum_{i=1}^{n}\sum_{k=1}^{n}b_{ik}a_{ki}\), which are equal by changing the order of summation, proving \(\operatorname{Tr}(AB) = \operatorname{Tr}(BA)\).
04

Using Traces to Deduce Impossibility

Assuming \(AB - BA = I_n\), we take the trace to get \(\operatorname{Tr}(AB - BA) = \operatorname{Tr}(I_n)\). This implies \( \operatorname{Tr}(AB) - \operatorname{Tr}(BA) = n \). However, since \(\operatorname{Tr}(AB) = \operatorname{Tr}(BA)\), we must have \(0 = n\), which is a contradiction. Therefore, no such matrices \(A\) and \(B\) can exist.

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Key Concepts

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

Matrix Addition
Matrix addition is a fundamental operation in linear algebra where each element of one matrix is added to the corresponding element of another matrix of the same dimension. For two matrices, say \( A = [a_{ij}] \) and \( B = [b_{ij}] \), the sum \( A + B \) is calculated element-wise:
  • \( (A + B)_{ij} = a_{ij} + b_{ij} \)

This operation is straightforward but immensely useful, especially since it preserves the dimensional consistency of the matrices. When discussing the trace related to matrix addition, it's important to remember the straightforward nature of how the sum of diagonal elements working similarly as simple addition of numbers.
Matrix Multiplication
Matrix multiplication involves more complexity than addition. When you multiply two matrices, like \( A = [a_{ij}] \) and \( B = [b_{ij}] \), you perform a dot product across the rows of the first matrix and the columns of the second:
  • The element at the \( i^{th} \) row and \( j^{th} \) column of the product \( AB \) is given by \((AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} \)

Matrix multiplication is not commutative, which means \( AB \) is not necessarily equal to \( BA \). Nevertheless, the operation is associative and distributive over matrix addition, making it a crucial tool in many mathematical computations.
Commutator of Matrices
In the context of matrices, a commutator is given by the expression \( AB - BA \). When two matrices commute, this expression equals zero: \( AB = BA \).
  • If \( AB - BA = 0 \), matrices \( A \) and \( B \) are said to commute.

Understanding commutators is essential as they provide insight into the symmetries and characteristics of linear transformations represented by matrices. In our context, utilizing the trace function helps determine that \( AB \) and \( BA \) share similar eigenvalue properties.
Diagonal Elements of a Matrix
The diagonal elements of a matrix are the components \( a_{ii} \) for which the row number equals the column number. In a matrix \( A = [a_{ij}] \), these elements form the diagonal from the top left to the bottom right.
  • The significance of diagonal elements is central in operations like calculating the trace, denoted by \( \operatorname{Tr}(A) \), which is the sum of these elements.

For example, in a 3x3 matrix:
  • If \( A = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{pmatrix} \), then the diagonal elements are \( 1, 5, \) and \( 9 \), making \( \operatorname{Tr}(A) = 1 + 5 + 9 = 15 \).

These diagonal sums are useful in a variety of calculations, especially those involving matrix operations like addition and multiplication.

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Most popular questions from this chapter

(a) Find \(2 \times 2\) matrices \(A\) and \(B\) such that \((A B)^{2} \neq A^{2} B^{2}\). (b) Find \(2 \times 2\) matrices \(C\) and \(D\) such that \(C D \neq D C\) and yet \((C D)^{2}\) is equal to \(C^{2} D^{2}\).

(a) The transpose of the \(m \times n\) matrix \(A\) is the \(n \times m\) matrix \(A^{T}\) whose \(i\) th row is the ith column of \(A\). Thus, for example, $$ \left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]^{T}=\left[\begin{array}{ll} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array}\right] $$ Prove that if \(A\) and \(B\) are \(m \times n\) matrices and if \(C\) is an \(n \times r\) matrix, then \(\left(A^{T}\right)^{T}=A\), \((A+B)^{T}=A^{T}+B^{T}\) and \((A C)^{T}=C^{T} A^{T}\). If \(m=n\), deduce, for all positive integers \(r\), that \(\left(A^{\prime}\right)^{T}=\left(A^{T}\right)^{r}\). (b) The matrix \(A\) is symmetric if and only if \(A^{T}=A\). Show that, for each square matrix \(B\), both \(B B^{T}\) and \(B+B^{T}\) are symmetric. Suppose that \(C\) and \(D\) are symmetric \(n \times n\) matrices. Show that \(C D\) is symmetric if and only if \(C D=D C\). (c) The \(n \times n\) matrix \(A\) is said to be orthogonal if and only if \(A A^{T}=A^{T} A=I_{n}\). Regarding the rows of \(A\) as (row) vectors \(\mathbf{r}_{1}, \mathbf{r}_{2}, \ldots, \mathbf{r}_{n}\) and the columns of \(A\) as (column) vectors \(\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\) show (i) that \(A\) is orthogonal if and only if \(\mathbf{r}_{i} \mathbf{r}_{j}=1\) if \(i=j\) and 0 if \(i \neq j\), a similar result being true for columns. Show, also, (ii) that \(A\) is orthogonal if and only if \(A \mathbf{v} . A \mathbf{v}=\mathbf{v} . \mathbf{v}\) for each \(^{d} \mathbf{v} \in \mathbb{R}^{n}\), that is, if and only if multiplication by \(A\) preserves 'lengths' of vectors in \(\mathbb{R}^{n}\). [Hint: \(\mathbf{v} \cdot \mathbf{v}=\mathbf{v}^{T} I_{n} \mathbf{v}\).] What is the geometric effect of the mapping \(\mathrm{v} \rightarrow A \mathbf{v}\) if \(n=2\) or if \(n=3 ?\)

Prove that if \(A\) is \(m \times n\) and if \(\alpha\) is a scalar, then \(\left(\alpha I_{m}\right) A=\alpha A\).

Find \(a, b, c\) and \(d\) such that $$ \left(\left[\begin{array}{cc} 3 & -1 \\ 5 & 2 \end{array}\right]+\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\right)\left[\begin{array}{cc} 9 & 4 \\ 6 & -2 \end{array}\right]=\left[\begin{array}{ll} 7 & 1 \\ 7 & 4 \end{array}\right] $$ Are \(a, b, c\) and \(d\) uniquely determined?

(a) Let $$ A=\left[\begin{array}{cc} 3 & -1 \\ 2 & 5 \end{array}\right] $$ Find (using only pencil and paper) \(A^{16}\) and \(A^{13}\). [Hint: \(A^{16}=\left(\left(\left(A^{2}\right)^{2}\right)^{2}\right)^{2}\) and \(A^{13}=A^{8} A^{4} \cdot A\). It is nice to get \(A^{16}\) with only four multiplications!] (b) Given that $$ \left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right]^{5039} $$ find \(a, b, c\) and \(d\). Prove your claim is correct by mathematical induction. [Hint: to see what the induction hypothesis should be, evaluate \(\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]^{n}\) for \(n=2,3,4 .\) ]
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