What does it mean to have two matrix variables written vertically inside parentheses?
In other words, if A and B are matrices, what does this mean (imagine there is one set of parentheses spanning both rows):
(A)
(B)
I'm trying to understand a least squares problem like this:
(A) * C = (D)
(B) (0)
I understand regular least squares problems with matrices, but the parenthesis notation is confusing me.
Thanks!
In the context of matrices, writing two matrix variables vertically inside parentheses is known as a block matrix or matrix partitioning. It is used to represent the combination of multiple matrices into a larger matrix.
In your example, (A) * C = (D)
(B) (0)
The vertical parentheses indicate that the matrices A and B are stacked on top of each other to form a single matrix.
The equation can be interpreted as follows:
- The matrix C is multiplied by the block matrix, which includes A and B.
- The resulting matrix after the multiplication is equal to the block matrix (D) in the first row and (0) in the second row.
Essentially, the block matrix notation allows you to work with multiple matrices within a single expression, helping to simplify calculations and represent relationships between matrices.
The notation you mentioned, with two matrix variables written vertically inside parentheses, represents a block matrix or matrix partitioning. It is a way to represent a system of linear equations or a matrix equation that involves multiple submatrices.
In your example, the block matrix representation is used to represent a least squares problem involving matrices A, B, C, and D. Let's break down the notation:
(A)
(B)
This vertically stacked notation indicates that the matrix is composed of two submatrices: A and B. In other words, the matrix A sits above the matrix B inside the parentheses.
Now let's look at the matrix equation:
(A) * C = (D)
(B) (0)
Here, (A) represents the matrix A, (B) represents the matrix B, (D) represents the matrix D, and (0) represents a zero matrix with the same dimensions as B.
The equation (A) * C = (D) represents a system of linear equations where the matrix A is multiplied by the matrix C, and the result is equal to the matrix D. Similarly, the equation (B) * C = (0) represents another system of linear equations where the matrix B is multiplied by the matrix C, and the result is the zero matrix.
By organizing the matrices in a block matrix form, it becomes easier to analyze and solve the system of equations or perform operations on the matrices. Each submatrix can be treated as a separate entity, and operations can be performed on them individually or collectively.
To understand and solve the least squares problem, you would typically use matrix operations such as matrix multiplication, transposition, and inversion. Once you understand the block matrix notation, you can proceed with solving the system of equations or performing any necessary computations.