Question

The trace of a square matrix \(A,\) denoted tr \(A\), is the sum of the elements on the main diagonal of A. Show that, if \(A\) and \(B\) are \(n \times n\) matrices: a. \(\operatorname{tr}(A+B)=\operatorname{tr} A+\operatorname{tr} B\). b. \(\operatorname{tr}(k A)=k \operatorname{tr}(A)\) for any number \(k\). c. \(\operatorname{tr}\left(A^{T}\right)=\operatorname{tr}(A)\). d. \(\operatorname{tr}(A B)=\operatorname{tr}(B A)\). e. \(\operatorname{tr}\left(A A^{T}\right)\) is the sum of the squares of all entries of \(A\).

Short answer

a) tr(A+B) = trA + trB; b) tr(kA) = k tr(A); c) tr(A^T) = tr(A); d) tr(AB) = tr(BA); e) tr(AA^T) is the sum of squares of all entries of A.

Step-by-step solution

Definition of the Trace of a Matrix

The trace of a square matrix, denoted as \( \text{tr} A \), is the sum of the diagonal elements of the matrix. If a matrix \( A \) is represented as \([a_{ij}]\), where \( i \) and \( j \) are index positions, the trace is given by \( \text{tr} A = \sum_{i=1}^n a_{ii} \).

Proving \( \text{tr}(A + B) = \text{tr} A + \text{tr} B \)

Given two \( n \times n \) matrices \( A \) and \( B \), their sum \( A + B \) is also an \( n \times n \) matrix whose diagonal elements are the sums of the respective diagonal elements of \( A \) and \( B \). Therefore, \( \text{tr}(A + B) = \sum_{i=1}^n (a_{ii} + b_{ii}) = \sum_{i=1}^n a_{ii} + \sum_{i=1}^n b_{ii} = \text{tr} A + \text{tr} B \).

Proving \( \text{tr}(kA) = k \text{tr} A \)

For a scalar \( k \) and a matrix \( A = [a_{ij}] \), the matrix \( kA \) is formed by multiplying each element of \( A \) by \( k \). So the diagonal elements of \( kA \) are \( ka_{ii} \). Thus, \( \text{tr}(kA) = \sum_{i=1}^n ka_{ii} = k \sum_{i=1}^n a_{ii} = k \text{tr} A \).

Proving \( \text{tr}(A^T) = \text{tr}(A) \)

The transpose of a matrix \( A \), denoted \( A^T \), has the property that its diagonal elements are the same as those of \( A \) (since transpose does not affect diagonal elements). Hence, \( \text{tr}(A^T) = \sum_{i=1}^n a_{ii} = \text{tr} A \).

Proving \( \text{tr}(AB) = \text{tr}(BA) \)

For two \( n \times n \) matrices \( A \) and \( B \), the (\( i,i \))th element of \( AB \) is \( \sum_{j=1}^n a_{ij} b_{ji} \), and for \( BA \) it is \( \sum_{j=1}^n b_{ij} a_{ji} \). Due to the properties of the trace and symmetry of the cyclic property, it results in \( \text{tr}(AB) = \text{tr}(BA) \).

Proving \( \text{tr}(A A^T) \) is the Sum of Squares of All Entries of \(A\)

For matrix \( A \), the product \( AA^T \) consists of elements where the \( (i, i) \) entry is \( \sum_{j=1}^n a_{ij} a_{ij} = \sum_{j=1}^n a_{ij}^2 \). The trace of \( A A^T \) is \( \sum_{i=1}^n \sum_{j=1}^n a_{ij}^2 \), which is the sum of squares of all entries in \( A \).

Key concepts

Matrix Addition

Matrix addition is a straightforward operation where two matrices of the same dimensions are added together. This is done by adding corresponding elements from each matrix. For instance, if you have two matrices, \( A \) and \( B \), both of size \( n \times n \), you add them element-wise.

• If \( A = [a_{ij}] \) and \( B = [b_{ij}] \), the matrix \( A + B \) will have elements \( (a_{ij} + b_{ij}) \).
• The resultant matrix will also be of size \( n \times n \).
• Matrix addition is both commutative, meaning \( A + B = B + A \), and associative, meaning \( (A + B) + C = A + (B + C) \).

Matrix addition is used in the exercise to demonstrate that the trace of \( A + B \) is equal to the trace of \( A \) plus the trace of \( B \). This is because the sum of diagonal entries from \( A + B \) is simply the sum of the diagonal entries of \( A \) and \( B \).
This property facilitates many simplifications in linear algebra.

Matrix Scalar Multiplication

Matrix scalar multiplication involves multiplying each element of a matrix by a scalar (a constant). Consider a matrix \( A = [a_{ij}] \) and a scalar \( k \).

• The resulting matrix \( kA \) is achieved by multiplying each element \( a_{ij} \) of \( A \) by \( k \).
• If \( A \) is an \( n \times n \) matrix, \( kA \) remains an \( n \times n \) matrix because multiplication by a scalar does not change the size.
• Scalar multiplication is distributive, meaning \( k(A + B) = kA + kB \), and associative with respect to scalar products, meaning \( (ab)A = a(bA) \).

In the context of traces, this property is very useful because it tells us that the trace of \( kA \) is \( k \) times the trace of \( A \). This concept simplifies many computations when dealing with linear transformations and their traces.

Matrix Transpose

The transpose of a matrix is created by swapping its rows with its columns. For a matrix \( A \) having elements \( [a_{ij}] \), the transpose \( A^T \) has elements \( [a_{ji}] \).

• The main diagonal, consisting of elements where the row index equals the column index, stays the same during transposition.
• Transpose is idempotent, meaning \( (A^T)^T = A \).
• For multiplication, the transpose of a product equals the product of transposes in reverse order: \( (AB)^T = B^T A^T \).

In the exercise, it is important to note that the trace of a matrix \( A \) and its transpose \( A^T \) remains the same. This is because the order of addition does not matter, and diagonals remain unchanged in a transpose.

Matrix Multiplication

Matrix multiplication is a bit trickier than addition or scalar multiplication. It involves a dot product of the rows of the first matrix with the columns of the second.

• For two matrices \( A \) of size \( m \times n \) and \( B \) of size \( n \times p \), the resulting product matrix \( AB \) has the dimensions \( m \times p \).
• The element at position \( (i, j) \) in the product is calculated as the sum of products of corresponding elements from the \( i \)-th row of \( A \) and the \( j \)-th column of \( B \).
• Matrix multiplication is associative, \( (AB)C = A(BC) \), but not commutative, meaning \( AB eq BA \) in general.

The exercise uses a crucial property of matrix multiplication for traces: \( \text{tr}(AB) = \text{tr}(BA) \). This is due to the cyclic nature of trace computation, which is invariant under cyclic permutations of matrices.