Maths for Signals and Systems Linear Algebra in Engineering DR TANIA STATHAKI Lectures 9, Friday 28th October 2016 READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON Determinants

The Determinant is a crucial number associated with square matrices only. It is denoted by . These are two different symbols we use for determinants. If a matrix is invertible, that means Furthermore, means that matrix is invertible. For a matrix the determinant is defined as . This formula is explicitly associated with the solution of the system where is a matrix. Properties of determinants cont. 1. . This is easy to show in the case of a matrix using the formula of the previous slide. 2. If we exchange two rows of a matrix the sign of the determinant reverses. Therefore:

If we perform an even number of row exchanges the determinant remains the same. If we perform an odd number of row exchanges the determinant changes sign. Hence, the determinant of a permutation matrix is or . and as expected. Properties of determinants cont. 3a. If a row is multiplied with a scalar, the determinant is multiplied with that scalar too, i.e., . 3b. Note that I observe linearity only for a single row.

4. Two equal rows leads to . As mentioned, if I exchange rows the sign of the determinant changes. In that case the matrix is the same and therefore, the determinant should remain the same. Therefore, the determinant must be zero. This is also expected from the fact that the matrix is not invertible. Properties of determinants cont. 5. Therefore, the determinant after elimination remains the same.

6. A row of zeros leads to . This can verified as follows for any matrix: 7. Consider an upper triangular matrix ( is a random element) I can easily show the above using the following steps: I transform the upper triangular matrix to a diagonal one using elimination. I use property 3a times. I end up with the determinant .

The same observations are valid for a lower triangular matrix. Properties of determinants cont. 8. when is singular. This is because if is singular I get a row of zeros by elimination. Using the same concept I can say that if is invertible then In general I have , product of pivots. 9. where and is a scalar. 10. . We use the decomposition of , . Therefore, . and have the same determinant due to the fact that they are triangular matrices

(so it is the product of the diagonal elements). The same observation is valid for and . This observation and the property yields . This property can also be proved by the use of induction. Determinant of a matrix The goal is to find the determinant of a matrix using the properties described previously. We know that and I can realize the above analysis for matrices. I break the determinant of a random matrix into 4 determinants of simpler matrices.

In the case of a matrix I break it into 27 determinants. And so on. Determinant of any matrix For the case of a matrix we got: The determinants which survive have strictly one entry from each row and each column.

The above is a universal conclusion. Determinant of any matrix cont. For the case of a matrix we got: As mentioned the determinants which survive have strictly one entry from each row and each column. Each of these determinants is obtained by the product its non-zero elements with a plus or a minus sign in front. The sign is determined by the number of reorderings required so that the matrix of interest becomes diagonal. For example

the second determinant shown above requires one re-ordering (swap of rows 2 and 3) to become the determinant of a diagonal matrix. One re-ordering implies, as shown previously, negating the sign of the original determinant. Determinant of any matrix cont. For the case of a matrix the determinant has 2 survived terms. For the case of a matrix the determinant has 6 survived terms.

For the case of a matrix the determinant has 24 survived terms. For the case of a matrix the determinant has survived terms. The elements from the first row can be chosen in different ways. The elements from the second row can be chosen in different ways. and so on Problem Find the determinant of the following matrix: (The answer is 0). The general form for the determinant For the case of a matrix the determinant has terms.

are different columns. In the above summation, half of the terms have a plus and half of them have a minus sign. The general form for the determinant cont. For the case of a matrix, the method of cofactors is a technique which helps us to connect a determinant to determinants of smaller matrices. For a matrix we have is the determinant of a matrix which is a sub-matrix of the original matrix.

Cofactors The cofactor of element is defined as follows: is the that is obtained from the original matrix if row and column are eliminated. We keep the if is even. We keep the if is odd. Cofactor formula along row 1: Generalization: Cofactor formula along row : Cofactor formula along column : Cofactor formula along any row or column can be used for the final estimation of the determinant.

Estimation of the inverse using cofactors For a matrix it is quite easy to show that The general formula for the inverse is given by: is the cofactor of which is a sum of products of entries. In general Solve when is square and invertible The solution of the system when is square and invertible can be now obtained from

Lets find the first element of vector . This is: We see that in the cofactors used are the same ones that we use when we calculate the determinant of using its first column. Therefore, with is obtained by if we replace the first column with In general, is obtained by if we replace the th column with . Solve when is square and invertible cont. Cramers rule: The solution of the system with is given by ,. is obtained by if we replace the th column with .

In practice we must find determinants. Geometrical interpretation of the determinant Consider to be a matrix of size . It can be proven that the determinant of is the area of the parallelogram with the column vectors of as the two of its sides. Consider to be a matrix of size . It can be proven that the determinant of is the volume of the parallelepiped with the column vectors of as the three of its sides. This observation is extended to matrices of dimension

where the determinants are parallelotopes. Geometrical interpretation of the determinant cont. Take . Then the parallelepiped mentioned previously is the unit cube and its volume is 1. Problem: Consider an orthogonal square matrix . Prove that or . Solution: But Take and double one of its vectors.

The determinant doubles as well (property 3a). In that case, the cubes volume doubles, i.e., you have two cubes sitting on top of each other. You may download for free a demonstration by Wolfram that illustrates the above observations http://demonstrations.wolfram.com/DeterminantsSeenGeometrically/.