Multivariate Analysis Dr. Asmaa Ghalib Jabir
Description: Multivariate Analysis Dr. Asmaa Ghalib Jabir Lecture One Matrix Algebra Matrix Algebra The Matrix It is an ordered array of a set of observations as m rows and n columns, those observations are called: elements. A(( aij )) , where
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slide1. Multivariate Analysis Dr. Asmaa Ghalib Jabir<br>
slide2. Lecture One Matrix Algebra<br>
slide3. Matrix Algebra The Matrix
It is an ordered array of a set of observations as m "rows" and n "columns", those observations are called: "elements".
A=(( aij )) , where is: i=1,2,3,…, m and j=1,2,3,…, n.
Trace of a Matrix
The sum of elements on the main diagonal of a square matrix (A) is called the "trace" of a matrix. For (n×n) matrix (A),<br>
slide4. Identity Matrix
The identity matrix is a square matrix with (1) one in each main diagonal position and zero elements. Identity matrix is a special case of a diagonal matrix.
The Vector
Vector is a n×1 matrix, that is, a matrix consisting of a single column of n elements.<br>
slide5. Diagonal Matrix
It is a square matrix with non-zero elements only on its main diagonal. While some of ai elements may be zero.
Triangular Matrix
Upper triangle Matrix
If a square matrix has non-zero elements only on and above its main diagonal, it is called "upper triangular" matrix.
Lower triangle Matrix
If a square matrix has non-zero elements only on and under its diagonal, it is called "lower triangular" matrix.
Null Matrix
The (n×m) null matrix has zero in each of its positions and denoted by (O).<br>
slide6. Addition and Subtraction of Matrices
The sum of two matrices of like dimensions is the matrix of the sum of corresponding elements. Let A & B are matrices of the same oreder (n×n), then: their addition will be:<br>
slide7. Multiplication of Matrix
It is necessary that the number of columns of matrix A be equal to the number of rows of matrix B. So, if A is of dimension p×r and B is of dimension r×q, then, the multiplication results is of p×q dimension with "ijth“ elements of C computed as:
In general,<br>
slide2. Lecture One Matrix Algebra<br>
slide3. Matrix Algebra The Matrix
It is an ordered array of a set of observations as m "rows" and n "columns", those observations are called: "elements".
A=(( aij )) , where is: i=1,2,3,…, m and j=1,2,3,…, n.
Trace of a Matrix
The sum of elements on the main diagonal of a square matrix (A) is called the "trace" of a matrix. For (n×n) matrix (A),<br>
slide4. Identity Matrix
The identity matrix is a square matrix with (1) one in each main diagonal position and zero elements. Identity matrix is a special case of a diagonal matrix.
The Vector
Vector is a n×1 matrix, that is, a matrix consisting of a single column of n elements.<br>
slide5. Diagonal Matrix
It is a square matrix with non-zero elements only on its main diagonal. While some of ai elements may be zero.
Triangular Matrix
Upper triangle Matrix
If a square matrix has non-zero elements only on and above its main diagonal, it is called "upper triangular" matrix.
Lower triangle Matrix
If a square matrix has non-zero elements only on and under its diagonal, it is called "lower triangular" matrix.
Null Matrix
The (n×m) null matrix has zero in each of its positions and denoted by (O).<br>
slide6. Addition and Subtraction of Matrices
The sum of two matrices of like dimensions is the matrix of the sum of corresponding elements. Let A & B are matrices of the same oreder (n×n), then: their addition will be:<br>
slide7. Multiplication of Matrix
It is necessary that the number of columns of matrix A be equal to the number of rows of matrix B. So, if A is of dimension p×r and B is of dimension r×q, then, the multiplication results is of p×q dimension with "ijth“ elements of C computed as:
In general,<br>