{"id":42119,"date":"2025-06-28T13:39:31","date_gmt":"2025-06-28T13:39:31","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=42119"},"modified":"2025-06-28T13:39:32","modified_gmt":"2025-06-28T13:39:32","slug":"state-whether-the-following-hessian-matrices-are-positive-definite-or-negative-definite","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/state-whether-the-following-hessian-matrices-are-positive-definite-or-negative-definite\/","title":{"rendered":"State whether the following Hessian matrices are positive definite or negative definite"},"content":{"rendered":"\n
. State whether the following Hessian matrices are positive definite or negative definite. Show your work<\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
To determine whether the following Hessian matrices are positive definite or negative definite, we use the method of checking the signs of their leading principal minors. A matrix is positive definite<\/strong> if all the leading principal minors are positive, and it is negative definite<\/strong> if the signs of the minors alternate starting with negative.<\/p>\n\n\n\n
Matrix (a):<\/h3>\n\n\n\n
H=[5\u22126\u22126\u22121423\u22126\u22124]H = \\begin{bmatrix} 5 & -6 & -6 \\\\ -1 & 4 & 2 \\\\ 3 & -6 & -4 \\end{bmatrix}H=\u200b5\u221213\u200b\u221264\u22126\u200b\u221262\u22124\u200b\u200b<\/p>\n\n\n\n
Step 1: First leading principal minor (1×1 matrix)<\/h4>\n\n\n\n
The first minor is simply the element in the top left corner, which is 5. Since 5 is positive, the first minor is positive. D1=5D_1 = 5D1\u200b=5<\/p>\n\n\n\n
Step 2: Second leading principal minor (2×2 matrix)<\/h4>\n\n\n\n
The second leading principal minor is the determinant of the top left 2×2 submatrix: D2=\u22235\u22126\u221214\u2223=(5)(4)\u2212(\u22126)(\u22121)=20\u22126=14D_2 = \\begin{vmatrix} 5 & -6 \\\\ -1 & 4 \\end{vmatrix} = (5)(4) – (-6)(-1) = 20 – 6 = 14D2\u200b=\u200b5\u22121\u200b\u221264\u200b\u200b=(5)(4)\u2212(\u22126)(\u22121)=20\u22126=14<\/p>\n\n\n\n
Since 14 is positive, the second minor is positive.<\/p>\n\n\n\n
Step 3: Third leading principal minor (3×3 matrix)<\/h4>\n\n\n\n
For the third minor, we calculate the determinant of the entire matrix: D3=\u22235\u22126\u22126\u22121423\u22126\u22124\u2223D_3 = \\begin{vmatrix} 5 & -6 & -6 \\\\ -1 & 4 & 2 \\\\ 3 & -6 & -4 \\end{vmatrix}D3\u200b=\u200b5\u221213\u200b\u221264\u22126\u200b\u221262\u22124\u200b\u200b<\/p>\n\n\n\n
After expanding the determinant, we find that: D3=4D_3 = 4D3\u200b=4<\/p>\n\n\n\n
Since 4 is positive, the third minor is also positive.<\/p>\n\n\n\n
Since all the leading minors are positive, the matrix is positive definite<\/strong>.<\/p>\n\n\n\n
\n\n\n\n
Matrix (b):<\/h3>\n\n\n\n
H=[\u22124\u22122\u22122\u22126]H = \\begin{bmatrix} -4 & -2 \\\\ -2 & -6 \\end{bmatrix}H=[\u22124\u22122\u200b\u22122\u22126\u200b]<\/p>\n\n\n\n
Step 1: First leading principal minor (1×1 matrix)<\/h4>\n\n\n\n
The first minor is simply the element in the top left corner, which is -4. Since -4 is negative, the first minor is negative. D1=\u22124D_1 = -4D1\u200b=\u22124<\/p>\n\n\n\n
Since the first leading principal minor is negative, we immediately conclude that the matrix is negative definite<\/strong> without needing to check the second minor.<\/p>\n\n\n\n
\n\n\n\n
Conclusion:<\/h3>\n\n\n\n
    \n
  • The matrix in part (a) is positive definite<\/strong> because all the leading principal minors are positive.<\/li>\n\n\n\n
  • The matrix in part (b) is negative definite<\/strong> because the first leading principal minor is negative<\/li>\n<\/ul>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
    . State whether the following Hessian matrices are positive definite or negative definite. Show your work The Correct Answer and Explanation is: To determine whether the following Hessian matrices are positive definite or negative definite, we use the method of checking the signs of their leading principal minors. A matrix is positive definite if all […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-42119","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42119","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=42119"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42119\/revisions"}],"predecessor-version":[{"id":42122,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42119\/revisions\/42122"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=42119"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=42119"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=42119"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}