@ankitgupta.1729 sir can you please tell this answer?

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samarpita
asked
in Linear Algebra
May 9

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1 vote

Which one of the following statements is NOT true for a square matrix A?

- If A is real symmetric, the eigen values of A are the diagonal elements of it.
- If all the principal minors of A are positive, all the eigen values of A are also positive.

My question is what is “principal minors of A” ?

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Best answer

If $A$ is $m \times n$ matrix, $I \subseteq \{1,2,..,m\},$ and $J \subseteq \{1,2,…,n\}.$ Both sets $I$ and $J$ have the same number of elements $k$ then $k \times k$ minor of $A$ is denoted as $[A]_{I,J}$ that corresponds to the rows with index in set $I$ and the columns with index in set $J$.

If we take $I = J$ then it becomes Principal Minor. Since it is a determinant, so, it is a scalar value.

Now, trying to explain with an example.

Suppose, we have a matrix $A_{4 \times 4}$ defines as $\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix}$

Then $I \subseteq \{1,2,3,4\}$ and $J \subseteq \{1,2,3,4\}$

Now, with $k=1,$ equal $I$ and $J$ are possible with $\{1\}, \{2\}, \{3\}, \{4\}$

Here, $\{1\}$ means $I = J = \{1\}$ and since $I$ corresponds to rows and $J$ corresponds to columns, So, we have to take element from first row and first column it means corresponding principal minor is $|a_{11}|$

Similarly, $\{2\}$ means $I = J = \{2\}$ and so, we have to take element from second row and second column corresponding principal minor is $|a_{22}|$

$\{3\}$ means $I = J = \{3\}$ and so, corresponding principal minor is $|a_{33}|$

$\{4\}$ means $I = J = \{4\}$ and so, corresponding principal minor is $|a_{44}|$

Now, with $k=2,$ equal $I$ and $J$ are possible with $\{1,2\}, \{1,3\}, \{1,4\},\{2,3\}, \{2,4\},\{3,4\}$

Here, $\{1,2\}$ means $I = J = \{1,2\}$ and since $I$ corresponds to rows and $J$ corresponds to columns, it means we have to take elements from first row and second row as well as first column and second column. So, corresponding principal minor is $\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 3^{rd} row, 4th row and 3^{rd} column, 4th column) (which are not in I and J sets)

Similarly, $\{1,3\}$ means $I = J = \{1,3\}$ and here, we have to take elements from first row and third row as well as first column and third column. So, corresponding principal minor is $\begin{vmatrix} a_{11} & a_{13} \\ a_{31} & a_{33} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 2nd row, 4th row and 2nd column, 4th column) (which are not in I and J sets)

$\{1,4\}$ means $I = J = \{1,4\}$ and here, we have to take elements from first row and fourth row as well as first column and fourth column. So, corresponding principal minor is $\begin{vmatrix} a_{11} & a_{14} \\ a_{41} & a_{44} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 2nd row, 3rd row and 2nd column, 3rd column) (which are not in I and J sets)

$\{2,3\}$ means $I = J = \{2,3\}$ and here, we have to take elements from second row and third row as well as second column and third column. So, corresponding principal minor is $\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 1st row, 4th row and 1st column, 4th column) (which are not in I and J sets)

$\{2,4\}$ means $I = J = \{2,4\}$ and here, we have to take elements from second row and fourth row as well as second column and fourth column. So, corresponding principal minor is $\begin{vmatrix} a_{22} & a_{24} \\ a_{42} & a_{44} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 1st row, 3rd row and 1st column, 3rd column) (which are not in I and J sets)

$\{3,4\}$ means $I = J = \{3,4\}$ and here, we have to take elements from third row and fourth row as well as third column and fourth column. So, corresponding principal minor is $\begin{vmatrix} a_{33} & a_{34} \\ a_{43} & a_{44} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 1st row, 2nd row and 1st column, 2nd column) (which are not in I and J sets)

Now, with $k=3,$ equal $I$ and $J$ are possible with $\{1,2,3\}, \{1,2,4\}, \{1,3,4\}, \{2,3,4\}$

Here, $\{1,2,3\}$ means $I = J = \{1,2,3\}$ and since $I$ corresponds to rows and $J$ corresponds to columns, it means we have to take elements from first row, second row and third row as well as first column, second column and third column. So, corresponding principal minor is $\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 4^{th} row and 4^{th} column) (which are not in I and J sets)

$\{1,2,4\}$ means $I = J = \{1,2,4\}$ and since $I$ corresponds to rows and $J$ corresponds to columns, it means we have to take elements from first row, second row and fourth row as well as first column, second column and fourth column. So, corresponding principal minor is $\begin{vmatrix} a_{11} & a_{12} & a_{14} \\ a_{21} & a_{22} & a_{24} \\ a_{41} & a_{42} & a_{44} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 3rd row and 3rd column) (which are not in I and J sets)

$\{1,3,4\}$ means $I = J = \{1,3,4\}$ and since $I$ corresponds to rows and $J$ corresponds to columns, it means we have to take elements from first row, third row and fourth row as well as first column, third column and fourth column. So, corresponding principal minor is $\begin{vmatrix} a_{11} & a_{13} & a_{14} \\ a_{31} & a_{33} & a_{34} \\ a_{41} & a_{43} & a_{44} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 2nd row and 2nd column) (which are not in I and J sets)

$\{2,3,4\}$ means $I = J = \{2,3,4\}$ and since $I$ corresponds to rows and $J$ corresponds to columns, it means we have to take elements from second row, third row and fourth row as well as second column, third column and fourth column. So, corresponding principal minor is $\begin{vmatrix} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \\ \end{vmatrix}$

(We can think it as determinant of submatrix after removing 1st row and 1st column) (which are not in I and J sets)

Now, with $k=4,$ equal $I$ and $J$ are possible with $\{1,2,3,4\}$ which we have to take all the elements of the matrix, it means corresponding principal minor is $|A|$

Here. Both options are not true. Counter-Examples for each are:

$A)$ $\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}.$ Eigen values are $2,0$ which are not diagonal elements.

$B)$ $\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$. Here, principal minors are $1,1,2,$ all are positive but eigen values are $1+i$ and $1- i$ which are complex numbers and complex numbers are neither positive nor negative.