Question

If \(A\) and \(B\) are \(m \times n\) matrices, show that \(\operatorname{rank}(A+B) \leq \operatorname{rank} A+\operatorname{rank} B\).

Short answer

The rank of \(A + B\) is at most the sum of the ranks of \(A\) and \(B\).

Step-by-step solution

Understand the Problem

We need to show that the rank of the sum of two matrices \(A\) and \(B\) is less than or equal to the sum of their individual ranks, i.e., \(\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)\). This involves understanding the properties of matrix ranks and linear transformations.

Recall Rank Properties

The rank of a matrix is the dimension of its column space. For any two matrices \(A\) and \(B\) of the same size, the column space of \(A + B\) is contained within the sum of the column spaces of \(A\) and \(B\).

Apply Rank Inequality

Since the column space of \(A + B\) is a subspace of the sum of the column spaces of \(A\) and \(B\), we have the inequality: \[\operatorname{rank}(A+B) \leq \dim(\text{Col}(A) + \text{Col}(B)).\]

Use Subspace Dimension Formula

By the dimension formula for subspaces, we know: \[\dim(\text{Col}(A) + \text{Col}(B)) \leq \dim(\text{Col}(A)) + \dim(\text{Col}(B)).\] Thus, \(\dim(\text{Col}(A) + \text{Col}(B)) \leq \operatorname{rank}(A) + \operatorname{rank}(B)\).

Combining Results

Combining the inequalities, we have: \[\operatorname{rank}(A+B) \leq \dim(\text{Col}(A) + \text{Col}(B)) \leq \operatorname{rank}(A) + \operatorname{rank}(B).\] Thus, the statement \(\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)\) is proven.

Key concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra, often denoted by the symbol "+". When we add two matrices, say A and B, both must have the same number of rows and columns. This is because matrix addition is performed element-wise. Each element in matrix A is added to the corresponding element in matrix B.

• A and B must be of the same size, meaning they share the same dimensions, typically m × n where m is the number of rows and n is the number of columns.
• The resulting matrix, A+B, will also be an m × n matrix.
• If the matrices do not have the same dimension, their addition cannot be performed.

Understanding matrix addition is crucial when analyzing how operations affect the rank of matrices. Since the operation involves individually adding each element of the matrices, it means their properties such as dimensions remain unchanged. This forms the basis for exploring more complex concepts like rank inequality in matrix theory.

Column Space

The column space of a matrix, often denoted as Col(A) for a matrix A, is the set of all possible linear combinations of its columns. In simpler terms, the column space is the span of the columns of the matrix, and it represents all the vectors that can be obtained by multiplying the matrix by a vector.

• It is a subspace of the field's n-dimensional vector space, where n is the number of rows in the matrix.
• The column space provides insight into the solutions of the linear system associated with the matrix.

When dealing with matrix addition, the column space of the resulting matrix (A+B) is contained within the sum of the column spaces of A and B. This is a key point in the theory of matrix rank inequality. The rank of a matrix is the dimension of its column space, indicating how many linearly independent columns exist. Thus, understanding how column spaces behave under addition is essential for grasping the full meaning of rank-related inequalities.

Subspace Dimension

The concept of subspace dimension is integral to comprehending how spaces relate under various operations. The dimension of a subspace is determined by the number of linearly independent vectors it contains. This concept directly ties into matrix operations, as the dimension of the column space of a matrix is what we refer to as its rank.

• The rank of a matrix is equivalent to the dimension of its column space.
• When two matrices are added together, the dimension of the combined column space is critical to understanding rank inequalities.

According to the subspace dimension formula, the dimension of the sum of two subspaces will not exceed the sum of the dimensions of these individual subspaces. This is why when you add two matrices, the rank of their sum can only be as large as the sum of their ranks. This principle underpins many discussions around matrix rank inequality, highlighting how addition can affect the dimensions of subspaces involved, thus affecting overall ranks.