A matrix is said to be a semi-magic square if its row sums...

A matrix is said to be a semi-magic square if its row sums and column sums (i.e., the sum of entries in an individual row or column) all add up to the same number. An example is $\left(\begin{array}{lll}8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2\end{array}\right)$, whose row and column sums are all equal to 15 . (a) Explain why the set of all semi-magic squares is a subspace of the vector space of $3 \times 3$ matrices. (b) Prove that the $3 \times 3$ permutation matrices (1.30) span the space of semi-magic squares. What is its dimension? (c) A magic square also has the diagonal and anti-diagonal (running from top right to bottom left) add up to the common row and column sum; the preceding $3 \times 3$ example is magic. Does the set of $3 \times 3$ magic squares form a vector space? If so, what is its dimension? (d) Write down a formula for all $3 \times 3$ magic squares.

Key Concepts

Vector Space of Matrices

The set of all matrices of a given size, such as 3×3 matrices, forms a vector space because it satisfies the axioms of vector addition and scalar multiplication. This concept is foundational in linear algebra, where matrices can be added together and multiplied by scalars to form new matrices, allowing the structure to be analyzed using techniques from vector space theory.

Subspace

A subspace is a subset of a vector space that is itself a vector space under the same operations of addition and scalar multiplication. In the context of semi?magic or magic squares, the set is defined by additional linear conditions (e.g., equal row and column sums), and verifying these conditions shows that the subset is closed under addition and scalar multiplication, which is required for the structure to be a subspace.

Linear Constraints

Linear constraints are equations that restrict the entries of a matrix in a way that is linear. For semi?magic and magic squares, constraints such as having the row sums and column sums equal impose linear conditions on the matrix entries. These constraints can be used to determine the dimension of the subspace defined by these squares and to verify closure properties under linear operations.

Permutation Matrices

Permutation matrices are particular matrices with exactly one entry of 1 in each row and each column and 0s elsewhere. They serve as a useful basis or spanning set in problems involving magic or semi?magic squares because they capture combinatorial symmetries and can be used to express any matrix in the subspace as a linear combination of these well?structured matrices.

Span and Basis

The span of a set of matrices is the collection of all linear combinations of those matrices. When a set of matrices, such as the permutation matrices in this context, spans a space, it means that any matrix in the space can be expressed as a combination of these matrices. A basis is a minimal spanning set that is also linearly independent; determining a basis and the dimension of the space is central to understanding its structure.

Dimension

The dimension of a vector space or subspace is the number of vectors in a basis for that space. In problems like these, linear constraints reduce the number of free variables, which in turn determines the dimension of the subspace of semi-magic or magic squares. Calculating the dimension helps in characterizing the subspace and understanding the degrees of freedom available within it.

Magic Squares versus Semi-Magic Squares

A magic square not only has rows and columns that sum to a common constant but also requires that the sums of the main diagonal and the anti-diagonal match that same constant. By adding these extra constraints, the set of magic squares becomes a more restricted subspace compared to semi?magic squares, and analyzing these additional conditions is crucial to determining the structure and dimension of the space of magic squares.

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