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Chapter 8: Problem 17

Let \(A\) be a real symmetrie matrix \(\neq 0 .\) Show that \(\operatorname{tr}(A A)>0 .\)

Short Answer

Expert verified
Given a nonzero real symmetric matrix A, we have A = A^T. Let B = AA, which can be rewritten as B = AA^T. To calculate tr(AA), we find the elements of B: b_{ij} = \(\sum_{k=1}^n a_{ik}a_{kj}\). Recall that the trace is the sum of the diagonal elements of B, so tr(B) = \(\sum_{i=1}^n \sum_{k=1}^n a_{ik}a_{ki}\). This can be rewritten as tr(B) = \(\sum_{i=1}^n \sum_{k=1}^n (a_{ik})^2\), which is strictly greater than 0 because there is at least one nonzero element a_{pq} and \((a_{pq})^2 > 0\). Thus, tr(AA) > 0 for any nonzero real symmetric matrix A.

Step by step solution

01

Rewrite the matrix product given the matrix is symmetric

Since A is symmetric, we have A = A^T. Let's denote the matrix product AA as B, so B = AA. We can rewrite B as a product of A and its transpose: B = AA^T.
02

Calculate the elements in B

To show that tr(AA) > 0, we need to calculate the elements of B. For any matrix C, if C = AB, then c_{ij} = \(\sum_{k=1}^n a_{ik}b_{kj}\). In our case, since B = AA^T, we have b_{ij} = \(\sum_{k=1}^n a_{ik}a_{kj}\).
03

Calculate the trace of B

The trace of a matrix is the sum of its diagonal elements. Therefore, tr(B) = \(\sum_{i=1}^n b_{ii}\). Using the formula for b_{ij} from Step 2, we have tr(B) = \(\sum_{i=1}^n \sum_{k=1}^n a_{ik}a_{ki}\).
04

Rewrite tr(B) and show tr(B) > 0

We can rewrite tr(B) = \(\sum_{i=1}^n \sum_{k=1}^n a_{ik}a_{ki}\) as tr(B) = \(\sum_{i=1}^n \sum_{k=1}^n (a_{ik})^2\), since A is symmetric and thus A^T = A. Notice that \((a_{ik})^2\) is always non-negative, since the square of any real number is non-negative. Moreover, A is a nonzero matrix, which means there is at least one element a_{pq} such that a_{pq} 鈮� 0, in which case \((a_{pq})^2 > 0\), otherwise the sum would be zero. Therefore, tr(B) = \(\sum_{i=1}^n \sum_{k=1}^n (a_{ik})^2\) is strictly greater than 0. Hence, we have shown that the trace of the matrix product AA, tr(AA), is strictly greater than 0 for any nonzero real symmetric matrix A.

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Key Concepts

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

Symmetric Matrix
In the realm of linear algebra, matrices that exhibit symmetry are termed "symmetric matrices." A matrix is symmetric if it is equal to its transpose. In other words, a matrix \( A \) is symmetric if \( A = A^T \). This means that the elements above and below the main diagonal are mirror images of each other.
  • For a matrix \( A \) with elements \( a_{ij} \), the condition of symmetry implies \( a_{ij} = a_{ji} \) for all \( i, j \).
  • Symmetric matrices can only be square matrices, meaning they must have the same number of rows and columns.
  • Such matrices are crucial because they often appear in real-world applications, especially where stability and consistency are necessary.
This symmetric property plays a pivotal role in simplifying matrix operations, as seen in matrix equations and products. In the exercise, when solving the matrix product \( AA \) for a symmetric matrix \( A \), knowing that \( A = A^T \) provides a direct way to express the matrix product as \( AA^T \). This symmetry ensures certain mathematical traits, such as each eigenvalue being real.
Matrix Trace
The trace of a matrix is a straightforward yet significant concept in linear algebra. It is calculated as the sum of all the diagonal elements of a given square matrix. For a matrix \( A \) with diagonal elements \( a_{11}, a_{22}, ..., a_{nn} \), the trace is expressed as:
\[\text{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}\]
  • The trace is only defined for square matrices since it requires a well-defined diagonal.
  • This simple sum finds relevance in various fields, including statistics, physics, and in the solution of differential equations.
  • It is useful for determining properties like matrix invariants and is invariant under change of basis.
When considering the exercise, finding \( \text{tr}(AA) \) for the symmetric matrix relates to the sum of squares of all elements in \( A \), ensuring \( \text{tr}(AA) > 0 \) due to the non-zero nature of \( A \). This showcases how the trace can reflect specific properties of the matrix itself.
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to produce a third one. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. The resultant matrix's dimensions are determined by the rows of the first matrix and the columns of the second.
  • If matrices \( A \) (order \( m \times n \)) and \( B \) (order \( n \times p \)) are multiplied, the result is a matrix \( C \) of order \( m \times p \).
  • The element \( c_{ij} \) of the resulting matrix \( C \) is calculated by multiplying the elements of the \( i \)-th row of \( A \) with corresponding elements in the \( j \)-th column of \( B \) and summing them. This is given by \( c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} \).
In the given exercise, the matrix \( AA \) denotes the product of a matrix \( A \) with itself (or its transpose since it's symmetric). This operation is crucial as it reveals how symmetry impacts multiplication. Due to symmetry, the elements \( a_{ik} \) are mirrored across the diagonal, simplifying and ensuring non-negative diagonal sums when finding the trace.

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Most popular questions from this chapter

(a) Let \(V\) be a vector space over the field \(K\), and suppose that \(V\) is a direct sum, \(V=W_{1} \oplus W_{2}\), of subspaces \(W_{1}, W_{2}\). Let \(g_{1}\) be a symmetric bilinear form on \(W_{1}\), and let \(g_{2}\) be a symmetric bilinear form on \(W_{2}^{*}\) Show that there exists a unique bilinear form \(g\) on \(V\) such that, if \(v=v_{1}+v_{2}\) and \(w=w_{1}+w_{2}\) are elements of \(V\), with \(v_{1}, w_{1} \in \mathrm{II}_{1}\) and \(v_{2}, w_{2} \in W_{2}\), then \(g(v, w)=g_{1}\left(v_{1}, w_{1}\right)+g_{2}\left(v_{2}, w_{2}\right) .\) (b) If \(\mathbb{B}_{1}\) is a basis for \(W_{1}\) and \(\mathbb{B}_{2}\) is a basis for \(W_{2}\), what will the matrix of the form \(g\) look like, with respect to the basis \(\mathbb{B}_{1}, \mathbb{B}_{2}\) of \(V\) ?

Let \(V\) be a finite dimensional vector space over \(\mathbf{C}\), with s positive definite hermitian form. Let \(A: V \rightarrow V\) be a hermitian operator. Show that \(I+i A\) and \(I-i A\) are invertible.

Let \(V=\mathbf{C}\) be viewed as a vector space of dimension 2 over \(\mathbf{R}\). Let \(\alpha \in \mathbf{C}\), and let \(L_{a}: \mathbf{C} \rightarrow \mathbf{C}\) be the map \(z \mapsto \alpha z .\) Show that \(L_{a}\) is an \(\mathbf{R}\) -linear map of \(V\) into itself. For which complex numbers \(\alpha\) is \(L_{a}\) a unitary map with respect to the sealar product \(\langle z, w\rangle=\operatorname{Re}(z \vec{w}) ?\) What is the matrix of \(L_{a}\) with respect to the basis \(\\{1, i\\}\) of C over \(\mathbf{R}\) ?

Let \(V\) be a finite dimensional vector space over the field \(K\), with a nondegenerate bilinear form \(\langle,\),\(rangle . If A: V \rightarrow V\) is a linear map such that $$ \langle A v, A w\rangle=\langle v, w\rangle $$ for all \(v, w \in V\), show that \(\operatorname{Det}(A)=\pm 1\).

(a) Let \(V\) be a finite dimensional space over \(\mathbf{R}\), with a positive definite sealar product, and let \(\left\\{v_{1}, \ldots, v_{n}\right\\}=\beta\) and \(\left\\{w_{1}, \ldots, w_{n}\right\\}=B^{\prime}\) be orthonormal bases of \(V\). Show that the matrix \(M_{\mathbb{G}^{\prime}}^{\mathbb{W}}(i d)\) is real unitary. [Hint: Use \(\left\langle w_{i}, w_{i}\right\rangle=1\) and \(\left\langle w_{i}, w_{j}\right\rangle=0\) if \(i \not A j\), as well as the expression \(w_{i}=\sum a_{i j} v_{j}\), for some \(\left.a_{i j} \in \mathbf{R} .\right]\). (b) Let \(F: V \rightarrow V\) be such that \(F\left(v_{i}\right)=w_{i}\) for all \(i\). Show that \(M_{\text {' }}^{B}(F)\) is unitary.
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