This chapter is about breaking up a matrix into pieces that somehow combine to recreate . Usually the pieces are again matrices, and usually they are then combined via matrix multiplication (Definition MM). In some cases, the decomposition will be valid for any matrix, but often we might need extra conditions on , such as being square (Definition SQM) or diagonalizable (Definition DZM) before we can guarantee the decomposition. If you are comfortable with topics like decomposing a solution vector into linear combinations (Subsection LC.VFSS) or decomposing vector spaces into direct sums (Subsection PD.DS), then we will be doing similar things in this chapter. If not, review these ideas and take another look at Technique DC on decompositions.
We have studied one matrix decomposition already, so we will review that here in this introduction, both as a way of previewing the topic in a familiar setting, but also since it does not deserve another section all of its own.
A diagonalizable matrix (Definition DZM) is defined to be a square matrix such that there is an invertible matrix and a diagonal matrix where . Here we have a decomposition of into three matrices, , and , which recombine through matrix multiplication to recreate . We cannot form this decomposition for just any matrix — must be square and we know from Theorem DC that a matrix of size is diagonalizable if and only if there is a basis for composed entirely of eigenvectors of , or by Theorem DMFE we know that is diagonalizable if and only if each eigenvalue of has a geometric multiplicity equal to its algebraic multiplicity. Some authors prefer to call this an eigen decomposition of rather than a matrix diagonalization.
Each section of this chapter is a different matrix decomposition, and they will become progessively more general and more complicated. As such, later decompositions will require greater familiarity with topics from the Core (Part C).