A square matrix is called strictly lower (upper) triangular...

A square matrix is called strictly lower (upper) triangular if all entries on or above (below) the main diagonal are 0 . (a) Prove that every square matrix can be uniquely written as a sum $A=L+D+U$, with $L$ strictly lower triangular, $D$ diagonal, and $U$ strictly upper triangular. (b) Decompose $A=\left(\begin{array}{rrr}3 & 1 & -1 \\ 1 & -4 & 2 \\ -2 & 0 & 5\end{array}\right)$ in this manner.

A square matrix is called strictly lower (upper) triangular if all entries on or above (below) the main diagonal are 0 . (a) Prove that every square matrix can be uniquely written as a sum $A=L+D+U$, with $L$ strictly lower triangular, $D$ diagonal, and $U$ strictly upper triangular.
(b) Decompose $A=\left(\begin{array}{rrr}3 & 1 & -1 \\ 1 & -4 & 2 \\ -2 & 0 & 5\end{array}\right)$ in this manner.

Key Concepts

Matrix Decomposition

Matrix decomposition is a fundamental concept in linear algebra where a given matrix is expressed as a sum or product of matrices with simpler or more desirable structures. In this context, decomposing a square matrix into a sum of strictly lower triangular, diagonal, and strictly upper triangular matrices allows one to isolate and study different parts of the matrix with respect to their positions relative to the main diagonal. This breakdown can simplify understanding of the matrix's properties and is useful in algorithms and theoretical proofs.

Triangular Matrices

Triangular matrices are matrices that have zero entries either above or below the main diagonal. They come in two types: lower triangular matrices have zeros above the main diagonal, while upper triangular matrices have zeros below the main diagonal. These matrices are important in both computational methods and theoretical aspects of algebra because they simplify many matrix operations, such as solving systems of linear equations and calculating determinants.

Strictly Triangular Matrices

Strictly triangular matrices are a special type of triangular matrix where not only the entries above or below the main diagonal are zero, but the main diagonal itself is also zero. In the context of matrix decomposition, a strictly lower triangular matrix has zeros on and above the main diagonal, and a strictly upper triangular matrix has zeros on and below the main diagonal. This refinement is useful for ensuring the uniqueness of the decomposition when the diagonal component is separated out explicitly.

Diagonal Matrices

Diagonal matrices are matrices in which all off-diagonal entries are zero, meaning that only the entries on the main diagonal can be non-zero. They are significant in linear algebra due to their simple structure, which makes computations like matrix inversion, eigenvalue determination, and exponentiation more straightforward. In decompositions, the diagonal matrix captures the components that lie directly on the main diagonal of the original matrix.

Uniqueness of Decomposition

The uniqueness of the decomposition into strictly lower triangular, diagonal, and strictly upper triangular components is based on the fact that the position of each entry in a matrix uniquely belongs to one of these three subsets. Since each position in the matrix can only be part of one of the specific matrices in the decomposition (depending on whether it is below, on, or above the main diagonal), the splitting process is unambiguous and leads to a unique representation.

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