Question

A symmetric matrix is one for which the transpose of the matrix is the same as the original matrix, \(A^{T}=A\). An antisymmetric matrix is one that satisfies \(A^{T}=-A\). a. Show that the diagonal elements of an \(n \times n\) antisymmetric matrix are all zero. b. Show that a general \(3 \times 3\) antisymmetric matrix has three independent off-diagonal elements. c. How many independent elements does a general \(3 \times 3\) symmetric matrix have? d. How many independent elements does a general \(n \times n\) symmetric matrix have? e. How many independent elements does a general \(n \times n\) antisymmetric matrix have?

Short answer

a. All diagonal elements of an antisymmetric matrix are zero. b. A 3x3 antisymmetric matrix has 3 independent off-diagonal elements. c. A 3x3 symmetric matrix has 6 independent elements. d. An nxn symmetric matrix has \(n(n+1)/2\) independent elements. e. An nxn antisymmetric matrix has \(n(n-1)/2\) independent elements.

Step-by-step solution

Diagonal Elements of Antisymmetric Matrix

The task is to show that the diagonal elements of an \(n \times n\) antisymmetric matrix are all zero. For any matrix A, the element at the intersection of the i-th row and the j-th column is denoted by \(A_{ij}\). Now, for an antisymmetric matrix, by definition, \(A^{T} = -A\). Hence, \(A_{ij} = -A_{ji}\). For diagonal elements, i = j, so \(A_{ii} = -A_{ii}\). The only number that equals its own negative is zero. So, all diagonal elements of an antisymmetric matrix are zero.

Independent Off-diagonal Elements of 3x3 Antisymmetric Matrix

To show that a general 3x3 antisymmetric matrix has 3 independent off-diagonal elements, it's necessary to construct the general form of such a matrix. A 3x3 antisymmetric matrix A would look like: \[\[0, a, b\], [-a, 0, c\], [-b, -c, 0]\]. Here, 'a', 'b', and 'c' are the independent non-diagonal elements. As it can be seen, there are 3 independent off-diagonal elements.

Independent Elements of 3x3 Symmetric Matrix

Symmetric matrices have the property that \(A_{ij}=A_{ji}\). Therefore, in a 3x3 symmetric matrix, the elements below the diagonal are the same as the elements above the diagonal. Hence, there are 3 diagonal elements and 3 independent pairs of off-diagonal elements, leading to a total of 6 independent elements in a 3x3 symmetric matrix.

Independent Elements of nxn Symmetric Matrix

In an \(n \times n\) symmetric matrix, each row (or column) has i independent elements, where i is the row (or column) number. The total number of independent elements would be the sum of independent elements in each row/column, which would sum up to \(n(n+1)/2\).

Independent Elements of nxn Antisymmetric Matrix

Similar to the symmetric matrix, an \(n \times n\) antisymmetric matrix has elements below the diagonal are negative of elements above the diagonal. However, unlike the symmetric matrix, diagonal elements must be zero. So, there are \(n(n-1)/2\) independent off-diagonal elements, as only half of the off-diagonal elements are distinct.

Key concepts

Symmetric Matrix

A symmetric matrix is a special type of square matrix that remains unchanged when transposed. This means that the elements of the matrix satisfy the condition \(A^{T} = A\), where \(A^{T}\) represents the transpose of matrix \(A\). In practical terms, this means that the elements that are mirrored over the main diagonal are equal.
For example, in a matrix:

• If \(A_{ij} = A_{ji}\), elements are symmetric around the diagonal.

This property leads to some interesting characteristics, particularly in relation to the number of independent elements, which will be discussed later. In symmetric matrices, the structure simplifies many mathematical computations and is widely used in various scientific fields.

Antisymmetric Matrix

An antisymmetric matrix is defined by a unique property. This type of matrix satisfies the condition \(A^{T} = -A\). Essentially,

• For any element \(A_{ij}\), it holds that \(A_{ij} = -A_{ji}\).

One key outcome of this property is that all diagonal elements must be zero. Here's why:
When \(i = j\), the equation becomes \(A_{ii} = -A_{ii}\). The only solution to this equation is zero, hence all diagonal positions \(A_{ii}\) in an antisymmetric matrix are zero. This feature of antisymmetric matrices is crucial when determining their independent elements.

Independent Elements

Independent elements are those which can be freely chosen without affecting the predetermined properties of the matrix. In symmetric matrices, the elements across the diagonal line are mirrored, meaning fewer independent elements are needed to define the whole matrix.
In a \(3 \times 3\) symmetric matrix, only 6 elements need to be independently selected: 3 diagonal and 3 off-diagonal elements. In contrast, a \(3 \times 3\) antisymmetric matrix has no independent diagonal elements and 3 independent off-diagonal elements, because each pair of mirror-image elements must be equal and opposite.
The concept of independent elements aids in understanding the underlying constraints and structure of the matrices, allowing a more efficient computation and usage of data.

3x3 Matrix

A \(3 \times 3\) matrix is a grid consisting of 3 rows and 3 columns, making up a total of 9 elements. These matrices are pivotal in matrix theory because they provide a compact way to analyze linear transformations in three-dimensional space.
In the context of symmetric and antisymmetric matrices, understanding the arrangement of elements is crucial.
For symmetric matrices, the relationships \(A_{ij} = A_{ji}\) imply that elements reflect symmetrically across the diagonal. For antisymmetric matrices, the relationships \(A_{ij} = -A_{ji}\) mean that off-diagonal elements must be opposites, and diagonal ones are zeros. Consequently, a 3x3 symmetric matrix has 6 and a 3x3 antisymmetric matrix has 3 independent elements, dictating how these matrices are constructed and interpreted.

Diagonal Elements

Diagonal elements of a matrix are those that reside on the line extending from the top left corner to the bottom right corner of a square matrix. In other words, they are the elements \(A_{ii}\) where the row index \(i\) equals the column index \(j\).
For symmetric matrices, these diagonal elements are independent and their freedom of choice contributes to defining the matrix, alongside a handful of off-diagonal pairs.
Conversely, for antisymmetric matrices, diagonal elements are fixed at zero to comply with the matrix's characteristic property \(A_{ii} = -A_{ii}\). Understanding the role of diagonal elements is vital to grasp how symmetrical properties influence the creation and constraints of such matrices.