What is the rank of the matrix?

The rank of a matrix is the maximum number of its linearly independent column vectors (or row vectors). From this definition it is obvious that the rank of a matrix cannot exceed the number of its rows (or columns).

What is block form of a matrix?

Review of block matrices A block matrix (also called partitioned matrix) is a matrix of the kind where , , and are matrices, called blocks, such that: and have the same number of rows; and have the same number of rows; and have the same number of columns; and have the same number of columns.

What is the rank of a lower triangular matrix?

The rank of the overall block triangular matrix is greater than or equal to the sum of the ranks of its diagonal blocks. I.e.: rank(A)≥rank(B)+rank(D). Equality is not necessarily attained, as exemplified by the following matrix [0010], which has rank 1 but diagonal blocks with rank 0.

What is the inverse of a block matrix?

Notice that the inverse of a block diagonal matrix is also block diagonal. Similarly, the inverse of a block secondary diagonal matrix is block secondary diagonal too, but in transposed partition so that there is a switch between B and C.

What is upper and lower triangular matrix?

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.

What is the rank of upper triangular matrix?

First, note that an upper triangular matrix has full rank if and only if the diagonal entries are non-zero. This is because the product of the diagonal entries is the determinant of the matrix. Now for each order matrix consider the order matrix formed by removing the first column and last row.